Linear Equations in Two Variables

Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations of a Line. Applications of Linear Equations and Graphing In this chapter we cover topics related to graphing and the applications of graphing. Graphs appear in magazines and newspapers and in other aspects of da-to-da life. Furthermore, in man fields of stud such as the sciences and business, graphs are used to displa data (information). The following words are ke terms used in this chapter. Search for them in the puzzle and in the tet throughout the chapter. B the end of this chapter, ou should be familiar with all of these terms. Ke Terms coordinate origin -ais -ais quadrant slope midpoint -intercept -intercept horizontal vertical W M V E P O L S W W M M R E N E F V D P D I D F K O O R Z J J P P Q Z E R G N V R P W L X J C K A W J O H R I F F X P P E B P F P Y X S J M V T I M T P E C R E T N I X C U R P N I W Y I N T E R C E P T R E W K N L C J V D Y Z R N G X D I T T D I V Y A Z L X N R T K A K V A N P G T E L L A Z T Q E H X V E N I R I N Y D K T T K N B T I Z Z I O W R A E H A N Y L K N A S Y X D P Q O R C E S O T X Q T S X K Q R D A V D L T O Z C U N F R Y O J O I E F A O U G I T D A I D A P X O M C H U L T K R R Z C E K X T T C Y K I Q R U R O D W L N B I S U Z W J U R R P S H E C X M N S N N 0

0 Chapter Linear Equations in Two Variables Section. Concepts. The Rectangular Coordinate Sstem. Plotting Points. The Midpoint Formula The Rectangular Coordinate Sstem and Midpoint Formula. The Rectangular Coordinate Sstem One application of algebra is the graphical representation of numerical information (or data). For eample, Table - shows the percentage of individuals who participate in leisure sports activities according to the age of the individual. Table - Percentage of Individuals Age Participating in Leisure (ears) Sports Activities 0 9% 0 % 0 % 0 % 60 % 70 8% Source: U.S. National Endowment for the Arts. Information in table form is difficult to picture and interpret. However, when the data are presented in a graph, there appears to be a downward trend in the participation in leisure sports activities for older age groups (Figure -). In this eample, two variables are related: age and the percentage of individuals who participate in leisure sports activities. Percent Percentage of Individuals Who Participate in Leisure Sports Activities Versus Age 70 60 0 0 0 0 0 0 0 0 0 0 0 0 60 70 80 Age of Participant (ears) Figure - To picture two variables simultaneousl, we use a graph with two number lines drawn at right angles to each other (Figure -). This forms a rectangular coordinate sstem. The horizontal line is called the -ais, and the vertical line is called the -ais. The point where the lines intersect is called the origin. On the -ais, the numbers to the right of the origin are positive, -ais 6 Quadrant II Quadrant I Origin 6 0 6 Quadrant III Quadrant IV 6 Figure - -ais

Section. The Rectangular Coordinate Sstem and Midpoint Formula 0 and the numbers to the left are negative. On the -ais, the numbers above the origin are positive, and the numbers below are negative. The - and -aes divide the graphing area into four regions called quadrants.. Plotting Points Points graphed in a rectangular coordinate sstem are defined b two numbers as an ordered pair (, ). The first number (called the first coordinate or abscissa) is the horizontal position from the origin. The second number (called the second coordinate or ordinate) is the vertical position from the origin. Eample shows how points are plotted in a rectangular coordinate sstem. Eample Plotting Points Plot each point and state the quadrant or ais where it is located. a. (, ) b. (, ) c. (, ) d. (, ) e. (0, ) f. (, 0) Solution: a. The point (, ) is in quadrant I. (, ) (0, ) (, ) (, 0) 0, (, ) Figure - b. The point (, ) is in quadrant. c. The point (, ) is in quadrant IV. d. The point (, ) can also be written as (., ). This point is in quadrant III. e. The point (0, ) is on the -ais. f. The point (, 0) is located on the -ais. TIP: Notice that the points (, ) and (, ) are in different quadrants. Changing the order of the coordinates changes the location of the point. That is wh points are represented b ordered pairs (Figure -). Skill Practice Plot the point and state the quadrant or ais where it is located. a. (, ) b. (, 0) c. (, ) d. (0, ) e. (, ) f. (, ) The effective use of graphs for mathematical models requires skill in identifing points and interpreting graphs. Eample Interpreting a Graph Kristine started a savings plan at the beginning of the ear and plotted the amount of mone she deposited in her savings account each month. The graph of her savings is shown in Figure -. The values on the -ais represent the first 6 months of the ear, and the values on the -ais represent the amount of mone in dollars that she saved. Refer to Figure - to answer the questions. Let represent Januar on the horizontal ais. Skill Practice Answers a. (, ); quadrant I b. (, 0); -ais c. (, ); quadrant IV d. (0, ); -ais e. (, ); quadrant III f. (, ); quadrant II (, ) (, 0) (, ) (0, ) 0 (, ) (, )

06 Chapter Linear Equations in Two Variables Savings ($) 00 80 60 0 0 0 0 6 Month Figure - TIP: The scale on the - and -aes ma be different. This often happens in applications. See Figure -. a. What is the -coordinate when the -coordinate is 6? Interpret the meaning of the ordered pair in the contet of this problem. b. In which month did she save the most? How much did she save? c. In which month did she save the least? How much did she save? d. How much did she save in March? e. In which two months did she save the same amount? How much did she save in these months? Solution: a. When is 6, the -coordinate is 0. This means that in June, Kristine saved $0. b. The point with the greatest -coordinate occurs when is. She saved the most mone, $90, in Februar. c. The point with the lowest -coordinate occurs when is. She saved the least amount, $0, in April. d. In March, the -coordinate is and the -coordinate is 80. She saved $80 in March. e. The two points with the same -coordinate occur when and when. She saved $60 in both Januar and Ma. Skill Practice Refer to Figure -. a. In which month(s) did Kristine save $80? b. How much did Kristine save in June? c. What was the total amount saved during these 6 months? Skill Practice Answers a. March b. $0 c. $0. The Midpoint Formula Consider two points in the coordinate plane and the line segment determined b the points. It is sometimes necessar to determine the point that is halfwa between the endpoints of the segment. This point is called the midpoint. If the coordinates of the (, ) endpoints are represented b, and,, then the midpoint of the segment is given b the following formula. Midpoint formula: a, b (, ) (, )

Section. The Rectangular Coordinate Sstem and Midpoint Formula 07 TIP: The midpoint of a line segment is found b taking the average of the -coordinates and the average of the -coordinates of the endpoints. Eample Finding the Midpoint of a Segment Find the midpoint of the line segment with the given endpoints. a., 6 and (8, ) b..,. and 6.6,. Solution: a., 6 and (8, ) a 8 a, 7 b, 6 b Appl the midpoint formula. Simplif. The midpoint of the segment is, 7. b..,. and 6.6,. a. 6.6,.9, 0.9.. b Appl the midpoint formula. Simplif. Skill Practice Find the midpoint of the line segment with the given endpoints.. (, 6) and ( 0, ). (.6, 6.) and (.,.) Eample Appling the Midpoint Formula A map of a national park is created so that the ranger station is at the origin of a rectangular grid. Two hikers are located at positions (, ) and, with respect to the ranger station, where all units are in miles. The hikers would like to meet at a point halfwa between them (Figure -), but the are too far apart to communicate their positions to each other via radio. However, the hikers are both within radio range of the ranger station. If the ranger station relas each hiker s position to the other, at what point on the map should the hikers meet? 6 (, ) Hiker 6 0 6 Ranger (, ) Station Hiker 6 Figure - Skill Practice Answers. a. 0.7,., b

08 Chapter Linear Equations in Two Variables Solution: To find the halfwa point on the line segment between the two hikers, appl the midpoint formula: (, ) and, and,, a, b a, b Appl the midpoint formula. a, b Simplif. The halfwa point between the hikers is located at, or., 0.. Skill Practice. Find the center of the circle in the figure, given that the endpoints of a diameter are (, ) and (7, 0). (, ) (7, 0) Skill Practice Answers., 6 Section. Boost our GRADE at mathzone.com! Stud Skills Eercises Practice Eercises Practice Problems Self-Tests NetTutor. After getting a test back, it is a good idea to correct the test so that ou do not make the same errors again. One recommended approach is to use a clean sheet of paper, and divide the paper down the middle verticall, as shown. For each problem that ou missed on the test, rework the problem correctl on the left-hand side of the paper. Then give a written eplanation on the right-hand side of the paper. To reinforce the correct procedure, return to the section of tet from which the problem was taken and do several more problems. Take the time this week to make corrections from our last test. e-professors Videos Perform the correct math here. () 0 Eplain the process here. Do multiplication before addition.

Section. The Rectangular Coordinate Sstem and Midpoint Formula 09. Define the ke terms. a. Rectangular coordinate sstem b. -Ais c. -Ais d. Origin e. Quadrant f. Ordered pair g. Midpoint Concept : The Rectangular Coordinate Sstem. Given the coordinates of a point, eplain how to determine which quadrant the point is in.. What is meant b the word ordered in the term ordered pair? Concept : Plotting Points. Plot the points on a rectangular coordinate sstem. a., b. 0, c. 0, 0 d., 0 e. a, 7 b f..,.7 6. Plot the points on a rectangular coordinate sstem. a., b. a, 0b c., d. 0, e., f., 7. A point on the -ais will have what -coordinate? 6 8. A point on the -ais will have what -coordinate? For Eercises 9, give the coordinates of the labeled points, and state the quadrant or ais where the point is located. 9. 0. A B C E D E A C B D

0 Chapter Linear Equations in Two Variables.. A B E C D A B 0 D E C For Eercises, refer to the graphs to answer the questions.. The fact that obesit is increasing in both children and adults is of great concern to health care providers. One wa to measure obesit is b using the bod mass inde. Bod mass is calculated based on the height and weight of an individual. The graph shows the relationship between bod mass inde and weight for a person who is 6 tall. Bod Mass Inde 0 0 0 0 0 8 66 7 8 90 98 06 Weight (lb) a. What is the bod mass inde for a 6 person who weighs lb? b. What is the weight of a 6 person whose bod mass inde is 9?. The graph shows the number of cases of West Nile virus reported in Colorado during the months of Ma through October 00. The month of Ma is represented b on the -ais. (Source: Centers for Disease Control.) Number of Cases 0 0 0 0 0 Number of Cases of West Nile Virus (Colorado, 00) 0 0 6 Month ( corresponds to Ma) a. Which month had the greatest number of cases reported? Approimatel how man cases were reported? b. Which month had the fewest cases reported? Approimatel how man cases were reported? c. Which months had fewer than 0 cases of the virus reported? d. Approimatel how man cases of the virus were reported in August?

Section. The Rectangular Coordinate Sstem and Midpoint Formula Concept : The Midpoint Formula For Eercises 8, find the midpoint of the line segment. Check our answers b plotting the midpoint on the graph.. 6. (, ) (, ) (0, ) (, ) 7. 8. (, ) (, ) (, ) (, ) For Eercises 9 6, find the midpoint of the line segment between the two given points. 9. (, 0) and ( 6, ) 0. ( 7, ) and,., 8 and,. 0, and,., and 6,. 9, and 0,..,. and.6,. 6. 0.8,. and., 7. 7. Two courier trucks leave the warehouse to make deliveries. One travels 0 mi north and 0 mi east. The other truck travels mi south and 0 mi east. If the two drivers want to meet for lunch at a restaurant at a point halfwa between them, where should the meet relative to the warehouse? (Hint: Label the warehouse as the origin, and find the coordinates of the restaurant. See the figure.) 0 mi 0 mi Restaurant Warehouse (0, 0) mi 0 mi

Chapter Linear Equations in Two Variables 8. A map of a hiking area is drawn so that the Visitor Center is at the origin of a rectangular grid. Two hikers are located at positions, and, with respect to the Visitor Center where all units are in miles. A campground is located eactl halfwa between the hikers. What are the coordinates of the campground? Hiker (, ) Visitor Center (0, 0) 0 Hiker (, ) 9. Find the center of a circle if a diameter of the circle has endpoints, and (, ). 0. Find the center of a circle if a diameter of the circle has endpoints, and 7,. Section. Concepts. Linear Equations in Two Variables. Graphing Linear Equations in Two Variables. -Intercepts and -Intercepts. Horizontal and Vertical Lines Linear Equations in Two Variables. Linear Equations in Two Variables Recall from Section. that an equation in the form a b 0 is called a linear equation in one variable. In this section we will stud linear equations in two variables. Linear Equation in Two Variables Let A, B, and C be real numbers such that A and B are not both zero. A linear equation in two variables is an equation that can be written in the form A B C This form is called standard form. A solution to a linear equation in two variables is an ordered pair, that makes the equation a true statement. Eample Determining Solutions to a Linear Equation For the linear equation 8, determine whether the order pair is a solution. a., 0 b., c. a, 0 b

Section. Linear Equations in Two Variables Solution: a. 8 The ordered pair, 0 indicates that and 0. 0? 8 Substitute and 0 into the equation. 8 0 8 (true) The ordered pair, 0 makes the equation a true statement. The ordered pair is a solution to the equation. b. 8 Test the point,.? 8 Substitute and into the equation.? 8 6? 8 (false) The ordered pair, does not make the equation a true statement. The ordered pair is not a solution to the equation. 0 c. 8 Test the point,. a 0 b? 8 Substitute and 0. Skill Practice 0 8 (true) The ordered pair, 0 is a solution to the equation. Determine whether each ordered pair is a solution for the equation 8. a., b., c.,.. Graphing Linear Equations in Two Variables Consider the linear equation. The solutions to the equation are ordered pairs such that the difference of and is. Several solutions are given in the following list: Solution Check (, ) (, 0) 0 (, ) (0, ) 0 ( ) (, ) ( ) (, ) ( ) Skill Practice Answers a. Not a solution b. Solution c. Solution

Chapter Linear Equations in Two Variables B graphing these ordered pairs, we see that the solution points line up (see Figure -6). There are actuall an infinite number of solutions to the equation. The graph of all solutions to a linear equation forms a line in the -plane. Conversel, each ordered pair on the line is a solution to the equation. To graph a linear equation, it is sufficient to find two solution points and draw the line between them. We will find three solution points and use the third point as a check point. This is demonstrated in Eample. (, ) (, 0) (, ) (, ) (0, ) Figure -6 Eample Graphing a Linear Equation in Two Variables Graph the equation. Solution: We will find three ordered pairs that are solutions to the equation. In the table, we have selected arbitrar values for or and must complete the ordered pairs. 0 (0, ) (,) (, ) From the first row, From the second row, From the third row, substitute 0. substitute. substitute. 0 0 0 0 The completed list of ordered pairs is shown as follows. To graph the equation, plot the three solutions and draw the line through the points (Figure -7). Arrows on the ends of the line indicate that points on the line etend infinitel in both directions. 0 (0, ) Q, R 0 (, 0) (0, ) (, ) (, 0) Figure -7

Section. Linear Equations in Two Variables Skill Practice. Given, complete the table and graph the line through the points. 0 Eample Graphing a Linear Equation in Two Variables Graph the equation. Solution: Because the -variable is isolated in the equation, it is eas to substitute a value for and simplif the right-hand side to find. Since an number for can be used, choose numbers that are multiples of that will simplif easil when multiplied b. 0 Substitute 0. Substitute. Substitute. 0 0 0 The completed list of ordered pairs is as follows. To graph the equation, plot the three solutions and draw the line through the points (Figure -8). Skill Practice 0 0,, 0 (, 0) (, 0) (0, ) (, ) Figure -8. Graph the equation Hint: Select values of that are multiples of.. Skill Practice Answers.. (, ) (, ) (0, )

6 Chapter Linear Equations in Two Variables. -Intercepts and -Intercepts For man applications of graphing, it is advantageous to know the points where a graph intersects the - or -ais. These points are called the - and -intercepts. In Figure -7, the -intercept is (, 0). In Figure -8, the -intercept is (, 0). In general, a point on the -ais must have a -coordinate of zero. In Figure -7, the -intercept is (0, ). In Figure -8, the -intercept is 0,. In general, a point on the -ais must have an -coordinate of zero. (a, 0) (0, b) Definition of - and -Intercepts An -intercept* is a point a, 0 where a graph intersects the -ais. (see Figure -9.) A -intercept is a point 0, b Figure -9.) where a graph intersects the -ais. (see Figure -9 *In some applications, an -intercept is defined as the -coordinate of a point of intersection that a graph makes with the -ais. For eample, if an -intercept is at the point (, 0), it is sometimes stated simpl as (the -coordinate is understood to be zero). Similarl,a -intercept is sometimes defined as the -coordinate of a point of intersection that a graph makes with the -ais. For eample, if a -intercept is at the point (0, 7), it ma be stated simpl as 7 (the -coordinate is understood to be zero). To find the - and -intercepts from an equation in and, follow these steps: Steps to Find the - and -Intercepts from an Equation Given an equation in and,. Find the -intercept(s) b substituting 0 into the equation and solving for.. Find the -intercept(s) b substituting 0 into the equation and solving for. Eample Finding the - and -Intercepts of a Line Find the - and -intercepts of the line 8. Then graph the line. Solution: To find the -intercept, substitute 0. 8 0 8 8 To find the -intercept, substitute 0. 8 0 8 8 The -intercept is (, 0). The -intercept is (0, ). In this case, the intercepts are two distinct points and ma be used to graph the line. A third point can be found to verif that the points all fall on the same line (points that lie on the same line are said to be collinear). Choose a different value for either or, such as.

Section. Linear Equations in Two Variables 7 8 8 6 8 8 Substitute. Solve for. The point (, ) lines up with the other two points (Figure -0). (, ) (0, ) (, 0) 8 Figure -0 Skill Practice. Given, find the - and -intercepts. Then graph the line. Eample Finding the - and -Intercepts of a Line Find the - and -intercepts of the line. Then graph the line. Solution: To find the -intercept, substitute 0. To find the -intercept, substitute 0. 0 0 The -intercept is (0, 0). The -intercept is (0, 0). Notice that the - and -intercepts are both located at the origin (0, 0). In this case, the intercepts do not ield two distinct points. Therefore, another point is necessar to draw the line. We ma pick an value for either or. However, for this equation, it would be particularl convenient to pick a value for that is a multiple of such as. 0 Substitute. The point (, ) is a solution to the equation (Figure -). Skill Practice 0 (, ) (0, 0) Figure -. Given, find the - and -intercepts. Then graph the line. Skill Practice Answers. (, 0) (0, ). (0, 0)

8 Chapter Linear Equations in Two Variables Eample 6 Interpreting the - and -Intercepts of a Line Companies and corporations are permitted to depreciate assets that have a known useful life span. This accounting practice is called straight-line depreciation. In this procedure the useful life span of the asset is determined, and then the asset is depreciated b an equal amount each ear until the taable value of the asset is equal to zero. The J. M. Gus trucking compan purchases a new truck for $6,000. The truck will be depreciated at $,000 per ear. The equation that describes the depreciation line is where represents the value of the truck in dollars and is the age of the truck in ears. a. Find the - and -intercepts. Plot the intercepts on a rectangular coordinate sstem, and draw the line that represents the straight-line depreciation. b. What does the -intercept represent in the contet of this problem? c. What does the -intercept represent in the contet of this problem? Solution: a. To find the -intercept, substitute To find the -intercept, substitute 0. 0. 0 6,000,000,000 6,000 6,000,000 6,000,0000 6,000 The -intercept is (, 0). The -intercept is (0, 6,000). Taable Value ($) Taable Value of a Truck Versus the Age of the Vehicle 70,000 60,000 0,000 0,000 0,000 0,000 0,000 0 0 6 Age (ears) b. The -intercept (, 0) indicates that when the truck is ears old, the taable value of the truck will be $0. c. The -intercept (0, 6,000) indicates that when the truck was new (0 ears old), its taable value was $6,000. Skill Practice Answers 6a. -intercept: (7., 0); -intercept: (0, 0) b. The -intercept (0, 0) represents the amount of fuel in the truck initiall (after 0 hr). After 0 hr, the tank contains 0 gal of fuel. c. The -intercept (7., 0) represents the amount of fuel in the truck after 7. hr. After 7. hr the tank contains 0 gal. It is empt. Skill Practice 6. Acme motor compan tests the engines of its trucks b running the engines in a laborator. The engines burn gal of fuel per hour. The engines begin the test with 0 gal of fuel. The equation 0 represents the amount of fuel left in the engine after hours. a. Find the - and -intercepts. b. Interpret the -intercept in the contet of this problem. c. Interpret the -intercept in the contet of this problem.

Section. Linear Equations in Two Variables 9. Horizontal and Vertical Lines Recall that a linear equation can be written in the form A B C, where A and B are not both zero. If either A or B is 0, then the resulting line is horizontal or vertical, respectivel. Definitions of Vertical and Horizontal Lines. A vertical line is a line that can be written in the form k, where k is a constant.. A horizontal line is a line that can be written in the form k, where k is a constant. Eample 7 Graphing a Vertical Line Graph the line 6. Solution: Because this equation is in the form k, the line is vertical and must cross the -ais at 6. We can also construct a table of solutions to the equation 6. The choice for the -coordinate must be 6, but can be an real number (Figure -). 6 8 6 6 6 8 6 8 6 6 8 8 6 6 8 Figure - Skill Practice 7. Graph the line. Eample 8 Graphing a Horizontal Line Graph the line 7. Solution: The equation 7 is equivalent to 7. Because the line is in the form k, the line must be horizontal and must pass through the -ais at 7 (Figure -). Skill Practice Answers 7.

0 Chapter Linear Equations in Two Variables We can also construct a table of solutions to the equation 7. The choice for the -coordinate must be, but can be an real number. 7 0 7 7 7 7 Figure - Skill Practice 8. Graph the line 9. Calculator Connections A viewing window of a graphing calculator shows a portion of a rectangular coordinate sstem. The standard viewing window for most calculators shows both the - and -aes between 0 and 0. Furthermore, the scale defined b the tick marks on both aes is usuall set to. The standard viewing window. Skill Practice Answers 8. 9 Linear equations can be analzed with a graphing calculator. It is important to isolate the -variable in the equation. Then enter the equation in the calculator. For eample, to enter the equation from Eample, we have: 8 A Table feature can be used to find man solutions to an equation. Several solutions to are shown here. 8 8

Section. Linear Equations in Two Variables A Graph feature can be used to graph a line. Sometimes the standard viewing window does not provide an adequate displa for the graph of an equation. For eample, in the standard viewing window, the graph of is visible onl in a small portion of the upper right corner. To see the - and -intercepts of this line, we can change the viewing window to accommodate larger values of and. Most calculators have a Range or Window feature that enables the user to change the minimum and maimum - and -values. In this case, we changed the values of to range between and 0, and the values of to range between 0 and 0. Section. Boost our GRADE at mathzone.com! Stud Skills Eercises Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. A good technique for studing for a test is to choose four problems from each section of the chapter and write each of them along with the directions on a card. On the back, put the page number where ou found that problem. Then shuffle the cards and test ourself on the procedure to solve each problem. For an that ou do not know how to solve, look at the page number and do several of that tpe. Write which four problems ou would choose for this section.. Define the ke terms. a. Linear equation in two variables b. -Intercept c. -Intercept d. Vertical line e. Horizontal line

Chapter Linear Equations in Two Variables Review Eercises. Plot each point on a rectangular coordinate sstem, and identif the quadrant or ais where it is located. a. A, b. B, c. C(, ) d. D0, For Eercises 6, find the midpoint of the line segment between the given points. Check our answer b graphing the line segment and midpoint.., and,. (7, 8) and, 6., 0 and, 0 Concept : Linear Equations in Two Variables For Eercises 7 0, determine if the ordered pair is a solution to the linear equation. 7. 9 8. 6 9. 0. a. 0, a. (0, ) a., 0 a. b. 6, b. a 6 b. (, ) b., 0b 0,, 7 c. a, 7 c., c. 6, c. b, Concept : Graphing Linear Equations in Two Variables For Eercises, complete the table. Then graph the line defined b the points... 6 0.. 0 0 6 6

Section. Linear Equations in Two Variables In Eercises 8, graph the linear equation.. 6. 8 7. 7 6 6 7 8 7 6 6 7 8 8. 9. 0. 6 7 6... 7 6.. 6.

Chapter Linear Equations in Two Variables 7. 8. Concept : -Intercepts and -Intercepts 9. Given a linear equation, how do ou find an -intercept? How do ou find a -intercept? 0. Can the point (, ) be an - or -intercept? Wh or wh not? For Eercises, a. find the -intercept, b. find the -intercept, and c. graph the line.. 8. 0. 8 7 6 6 7 8 9 6 7. 8. 6. 9 8 7 6 6 7 8 9

Section. Linear Equations in Two Variables 7. 8. 9. 0.... A salesperson makes a base salar of $0,000 a ear plus a % commission on the total sales for the ear. The earl salar can be epressed as a linear equation as 0,000 0.0 where represents the earl salar and represents the total earl sales. Total Yearl Salar ($) 60,000 0,000 0,000 Yearl Salar Relating to Sales 0 0 00,000 00,000 600,000 800,000 Total Yearl Sales ($) a. What is the salesperson s salar for a ear in which his sales total $00,000? b. What is the salesperson s salar for a ear in which his sales total $00,000? c. What does the -intercept mean in the contet of this problem? d. Wh is it unreasonable to use negative values for in this equation?

6 Chapter Linear Equations in Two Variables. A tai compan in Miami charges $.00 for an distance up to the first mile and $.0 for ever mile thereafter. The cost of a cab ride can be modeled graphicall. a. Eplain wh the first part of the model is represented b a horizontal line. b. What does the -intercept mean in the contet of this problem? c. Eplain wh the line representing the cost of traveling more than mi is not horizontal. d. How much would it cost to take a cab mi? Cost ($) Cost of Cab Ride Versus Number of Miles 6.00.00.00.00.00.00 0 0 Number of Miles Concept : Horizontal and Vertical Lines For Eercises, identif the line as either vertical or horizontal, and graph the line.. 6. 7. 8. 9. 6 0.. 9. 0

Section. Linear Equations in Two Variables 7 Epanding Your Skills For Eercises, find the - and -intercepts.... 7 a b Graphing Calculator Eercises For Eercises 6 9, solve the equation for. Use a graphing calculator to graph the equation on the standard viewing window. 6. 7 7. 8. 9 9. 0 0 For Eercises 60 6, use a graphing calculator to graph the lines on the suggested viewing window. 60. 6. 0 0 0 0 0 0 0 6. 6. 0. 0. 0. 0. For Eercises 6 6, graph the lines in parts (a) (c) on the same viewing window. Compare the graphs. Are the lines eactl the same? 6. a. 6. a. b.. b..9 c..9 c..

8 Chapter Linear Equations in Two Variables Section. Concepts. Introduction to the Slope of a Line. The Slope Formula. Parallel and Perpendicular Lines. Applications and Interpretation of Slope Slope of a Line. Introduction to the Slope of a Line In Section., we learned how to graph a linear equation and to identif its - and -intercepts. In this section, we learn about another important feature of a line called the slope of a line. Geometricall, slope measures the steepness of a line. Figure - shows a set of stairs with a wheelchair ramp to the side. Notice that the stairs are steeper than the ramp. ft ft ft Figure - 8 ft To measure the slope of a line quantitativel, consider two points on the line. The slope is the ratio of the vertical change between the two points to the horizontal change. That is, the slope is the ratio of the change in to the change in. As a memor device, we might think of the slope of a line as rise over run. Change in (run) Slope change in change in rise run B Change in (rise) To move from point A to point B on the stairs, rise ft and move to the right ft (Figure -). A -ft change in Figure - -ft change in Slope change in change in ft ft To move from point A to point B on the wheelchair ramp, rise ft and move to the right 8 ft (Figure -6). B Slope A change in change in ft 8 ft 6 The slope of the stairs is is 6. 8-ft change in Figure -6 -ft change in which is greater than the slope of the ramp, which

Section. Slope of a Line 9 Eample Finding the Slope in an Application Find the slope of the ladder against the wall. Solution: change in Slope change in ft ft ft or ft The slope is horizontall. Skill Practice which indicates that a person climbs ft for ever ft traveled. Find the slope of the roof. 8 ft 0 ft. The Slope Formula The slope of a line ma be found b using an two points on the line call these points (, ) and (, ). The change in between the points can be found b taking the difference of the -values:. The change in can be found b taking the difference of the -values in the same order:. (, ) Change in (, ) Change in The slope of a line is often smbolized b the letter m and is given b the following formula. Definition of the Slope of a Line The slope of a line passing through the distinct points (, ) and (, ) is m provided 0 Skill Practice Answers.

0 Chapter Linear Equations in Two Variables Eample Finding the Slope of a Line Through Two Points Find the slope of the line passing through the points (, ) and (7, ). Solution: To use the slope formula, first label the coordinates of each point, and then substitute their values into the slope formula., and 7,,, m 7 Label the points. Appl the slope formula. 6 Simplif. The slope of the line can be verified from the graph (Figure -7). (7, ) 6 6 7 8 9 0 (, ) Figure -7 TIP: The slope formula does not depend on which point is labeled (, ) and which point is labeled (, ). For eample, reversing the order in which the points are labeled in Eample results in the same slope:, and 7,,, then m 7 6 Skill Practice. Find the slope of the line that passes through the points (, ) and (6, 8). Skill Practice Answers. 0

Section. Slope of a Line When ou appl the slope formula, ou will see that the slope of a line ma be positive, negative, zero, or undefined. Lines that increase, or rise, from left to right have a positive slope. Lines that decrease, or fall, from left to right have a negative slope. Horizontal lines have a zero slope. Vertical lines have an undefined slope. Positive slope Negative slope Zero slope Undefined slope Eample Finding the Slope of a Line Between Two Points Find the slope of the line passing through the points (, ) and (, ). Solution:, and,, m 8 8, Label points. Appl the slope formula. Simplif. The two points can be graphed to verif that (Figure -8). 8 is the correct slope 8 (, ) (, ) The line slopes m 8 downward from left to right. Figure -8 Skill Practice. Find the slope of the line that passes through the given points. (, 8) and (, ) Skill Practice Answers.

Chapter Linear Equations in Two Variables Eample Finding the Slope of a Line Between Two Points (, ) (, ) Figure -9 a. Find the slope of the line passing through the points (, ) and (, ). b. Find the slope of the line passing through the points (0, ) and (, ). Solution: a., and,, m 6 6 0, Label points. Appl slope formula. Undefined The slope is undefined. The points define a vertical line (Figure -9). (, ) (0, ) Figure -0 b. 0, and,, 0 0 and m 0 Label the points. Appl the slope formula. Simplif. The slope is zero. The line through the two points is a horizontal line (Figure -0). Skill Practice, Find the slope of the line that passes through the given points.. (, ) and (, ). (, 6) and ( 7, 6). Parallel and Perpendicular Lines Lines in the same plane that do not intersect are parallel. Nonvertical parallel lines have the same slope and different -intercepts (Figure -). Lines that intersect at a right angle are perpendicular. If two lines are perpendicular, then the slope of one line is the opposite of the reciprocal of the slope of the other (provided neither line is vertical) (Figure -). Slopes of Parallel Lines m m If and represent the slopes of two parallel (nonvertical) lines, then See Figure -. m m Skill Practice Answers. Undefined. 0

Section. Slope of a Line These lines are perpendicular. These two lines are parallel. m m ft ft m 7 cm 7 cm 7 cm cm m 7 ft ft Figure - Figure - Slopes of Perpendicular Lines If m 0 and m 0 represent the slopes of two perpendicular lines, then m or equivalentl, m m. m See Figure -. Eample Determining the Slope of Parallel and Perpendicular Lines Suppose a given line has a slope of. a. Find the slope of a line parallel to the given line. b. Find the slope of a line perpendicular to the given line. Solution: a. The slope of a line parallel to the given line is m (same slope). b. The slope of a line perpendicular to the given line is m (the opposite of the reciprocal of ). Skill Practice L 6. The slope of line is. a. Find the slope of a line parallel to L. b. Find the slope of a line perpendicular to L. Eample 6 Determining Whether Two Lines Are Parallel, Perpendicular, or Neither Two points are given from each of two lines: L and L. Without graphing the points, determine if the lines are parallel, perpendicular, or neither. L :, and (, ) L :, 6 and, Skill Practice Answers 6a. b.

Chapter Linear Equations in Two Variables Solution: First determine the slope of each line. Then compare the values of the slopes to determine if the lines are parallel or perpendicular. For line : For line : L :, and (, ) L :, 6 and,,,,, Label the points. m 6 m Appl the slope formula. 8 The slope of L is. The slope of is L. The slope of L is the opposite of the reciprocal of L. B comparing the slopes, the lines must be perpendicular. Skill Practice 7. Two points are given for lines L and L. Determine if the lines are parallel, perpendicular, or neither. L : (, ) and (, 6) L : (, ) and (, 0). Applications and Interpretation of Slope Eample 7 Interpreting the Slope of a Line in an Application The number of males 0 ears old or older who were emploed full time in the United States varied linearl from 970 to 00. Approimatel.0 million males 0 ears old or older were emploed full time in 970. B 00, this number grew to 6. million (Figure -). Number (millions) Number of Males 0 Years or Older Emploed Full Time in the United States 70 (00, 6.) 60 0 0 (970,.0) 0 0 0 0 96 970 97 980 98 990 99 000 00 Figure - Source: Current population surve. a. Find the slope of the line, using the points (970,.0) and (00, 6.). b. Interpret the meaning of the slope in the contet of this problem. Skill Practice Answers 7. Parallel

Section. Slope of a Line Solution: a. 970,.0 and 00, 6.,, m 6..0 00 970 Label the points. Appl the slope formula. m. or m 0.6 b. The slope is approimatel 0.6, meaning that the full-time workforce has increased b approimatel 0.6 million men (or 60,000 men) per ear between 970 and 00. Skill Practice The number of people per square mile in Alaska was 0.96 in 990. This number increased to.7 in 00. 8a. Find the slope of the line that represents the population growth of Alaska. Use the points (990, 0.96) and (00,.7). b. Interpret the meaning of the slope in the contet of this problem. Skill Practice Answers 8a. 0.0 b. The population increased b 0.0 person per square mile per ear. Section. Boost our GRADE at mathzone.com! Stud Skills Eercises. Go to the online service called MathZone that accompanies this tet (www.mathzone.com). Name two features that this online service offers that can help ou in this course.. Define the ke term slope. Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos Review Eercises. Find the missing coordinate so that the ordered pairs are solutions to the equation a. 0, b.,0 c.,. For Eercises 7, find the - and -intercepts (if possible) for each equation, and sketch the graph.. 8 0. 0 6

6 Chapter Linear Equations in Two Variables 6. 6 0 7. 6 8 9 6 9 6 6 9 6 9 8 Concept : Introduction to the Slope of a Line 8. A -ft ladder is leaning against a house, as shown 9. Find the pitch (slope) of the roof in the figure. in the diagram. Find the slope of the ladder. 9 ft ft ft 0 ft 7 ft 0. Find the slope of the treadmill.. Find the average slope of the hill. 0 d 00 d 8 in. 7 in.. The road sign shown in the figure indicates the percent grade of a hill. This gives the slope of the road as the change in elevation per 00 horizontal ft. Given a % grade, write this as a slope in fractional form.. If a plane gains 000 ft in altitude over a distance of,000 horizontal ft, what is the slope? Eplain what this value means in the contet of the problem. % Grade Concept : The Slope Formula For Eercises 9, use the slope formula to determine the slope of the line containing the two points.. (6, 0) and 0,., 0 and 0, 6., and, 7. (, ) and, 0 8., and 7, 9., and, 0. 0.,. and 0., 0.8. 0., 0. and 0., 0.. (, ) and (, 7)

Section. Slope of a Line 7., and, 0., and,. 8, and (, ) 6..6,. and (0, 6.) 7. (., ) and., 0. 8. a and a 7, b, b 9. a and a, b 6, b 0. Eplain how to use the graph of a line to determine whether the slope of a line is positive, negative, zero, or undefined.. If the slope of a line is, how man units of change in will be produced b 6 units of change in? For Eercises 7, estimate the slope of the line from its graph..... 6. 7. Concept : Parallel and Perpendicular Lines 8. Can the slopes of two perpendicular lines both be positive? Eplain our answer.

8 Chapter Linear Equations in Two Variables For Eercises 9, the slope of a line is given. a. Find the slope of a line parallel to the given line. b. Find the slope of a line perpendicular to the given line. 9. m 0. m. m 7. m. m 0. m is undefined. In Eercises, two points are given from each of two lines L and L. Without graphing the points, determine if the lines are perpendicular, parallel, or neither.. L :, and, 9 6. L :, and, 7. L :, and, L :, and, L : 0, and 7, L :, and 0, 6 8. L : 0, 0 and, 9. L :, and, 9 0. L :, and L :, and 0, L :, and 0, L :, and, 0,. L :, and,. L : 7, and 0, 0 L :, and 0, L : 0, 8 and, 6 Concept : Applications and Interpretation of Slope. The graph shows the number of cellular phone subscriptions (in millions) purchased in the United States for selected ears. a. Use the coordinates of the given points to find the slope of the line, and epress the answer in decimal form. b. Interpret the meaning of the slope in the contet of this problem.. The number of SUVs (in millions) sold in the United States grew approimatel linearl between 990 and 00. a. Find the slope of the line defined b the two given points. b. Interpret the meaning of the slope in the contet of this problem. Millions Millions 0 00 0 00 0 Number of Cellular Phone Subscriptions (998, 70) (006, 0) 0 996 998 000 00 00 006 00 0 00 0 00 0 00 Number of SUVs Sold in the United States (99, ) (000, ) 0 0 990 99 99 996 998 000 00 00

Section. Slope of a Line 9. The data in the graph show the average weight for bos based on age. a. Use the coordinates of the given points to find the slope of the line. b. Interpret the meaning of the slope in the contet of this problem. Weight (lb) 00 80 60 0 0 Average Weight for Bos b Age (,.) Age (r) (0, 7.) 0 0 6 7 8 9 0 6. The data in the graph show the average weight for girls based on age. a. Use the coordinates of the given points to find the slope of the line, and write the answer in decimal form. b. Interpret the meaning of the slope in the contet of this problem. Weight (lb) Average Weight for Girls b Age 00 (, 87.) 90 80 70 60 0 0 (,.) 0 0 0 0 0 6 7 8 9 0 Age (r) Epanding Your Skills For Eercises 7 6, given a point P on a line and the slope m of the line, find a second point on the line (answers ma var). Hint: Graph the line to help ou find the second point. 7. P0, 0 and m 8. P, and m 9. P, and m is undefined 60. P, and m 0 6. P, and m 6. P, and m 7 6

0 Chapter Linear Equations in Two Variables Section. Concepts. Slope-Intercept Form. The Point-Slope Formula. Different Forms of Linear Equations Equations of a Line. Slope-Intercept Form In Section., we learned that an equation of the form A B C (where A and B are not both zero) represents a line in a rectangular coordinate sstem. An equation of a line written in this wa is said to be in standard form. In this section, we will learn a new form, called the slope-intercept form, which is useful in determining the slope and -intercept of a line. Let (0, b) represent the -intercept of a line. Let (, ) represent an other point on the line. Then the slope of the line through the two points is m m b m a b b m b S m b 0 m b or m b Appl the slope formula. Simplif. Clear fractions. Simplif. Solve for : slope-intercept form. Slope-Intercept Form of a Line m b is the slope-intercept form of a line. m is the slope and the point (0, b) is the -intercept. The equation 7 is written in slope-intercept form. B inspection, we can see that the slope of the line is and the -intercept is (0, 7). Eample Finding the Slope and -Intercept of a Line Given the line, write the equation of the line in slope-intercept form, then find the slope and -intercept. Solution: Write the equation in slope-intercept form, m b, b solving for. The slope is and the -intercept is (0, ). Skill Practice Skill Practice Answers. Slope: ; -intercept: a0, b. Write the equation in slope-intercept form. Determine the slope and the -intercept.

Section. Equations of a Line The slope-intercept form is a useful tool to graph a line. The -intercept is a known point on the line, and the slope indicates the direction of the line and can be used to find a second point. Using slope-intercept form to graph a line is demonstrated in Eample. Eample Graphing a Line b Using the Slope and -Intercept Graph the line b using the slope and -intercept. Solution: First plot the -intercept (0, ). The slope m can be written as m The change in is. The change in is. To find a second point on the line, start at the -intercept and move down units and to the right units. Then draw the line through the two points (Figure -). Similarl, the slope can be written as Start here -intercept (0, ) Figure - TIP: To graph a line using the -intercept and slope, alwas begin b plotting a point at the -intercept (not at the origin). m The change in is. The change in is. To find a second point on the line, start at the -intercept and move up units and to the left units. Then draw the line through the two points (see Figure -). Skill Practice. Graph the line b using the slope and -intercept. Two lines are parallel if the have the same slope and different -intercepts. Two lines are perpendicular if the slope of one line is the opposite of the reciprocal of the slope of the other line. Otherwise, the lines are neither parallel nor perpendicular. Eample Determining if Two Lines Are Parallel, Perpendicular, or Neither Given the pair of linear equations, determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular. a. L : 7 b. L : c. L : L : L : 6 L : 6 6 Skill Practice Answers.

Chapter Linear Equations in Two Variables Solution: a. The equations are written in slope-intercept form. L : L : The slope is and the -intercept is (0, 7). The slope is and the -intercept is 0,. Because the slopes are the same and the -intercepts are different, the lines are parallel. b. Write each equation in slope-intercept form b solving for. L : 7 Divide b. The slope of L is The slope of L is.. Divide b 6. The value is the opposite of the reciprocal of. Therefore, the lines are perpendicular. c. L : 6 is equivalent to 6. The slope is. L : L : 6 is a horizontal line, and the slope is 0. 6 6 The slopes are not the same. Therefore, the lines are not parallel. The slope of one line is not the opposite of the reciprocal of the other slope. Therefore, the lines are not perpendicular. The lines are neither parallel nor perpendicular. 6 6 6 6 Add to both sides. Skill Practice Given the pair of equations, determine if the lines are parallel, perpendicular, or neither.... 6 6 7 Eample Using Slope-Intercept Form to Find an Equation of a Line Use slope-intercept form to find an equation of a line with slope and passing through the point (, ). Skill Practice Answers. Perpendicular. Parallel. Neither Solution: To find an equation of a line in slope-intercept form, m b, it is necessar to find the slope, m, and the -intercept, b. The slope is given in the problem as m. Therefore, the slope-intercept form becomes m b b

Section. Equations of a Line Furthermore, because the point (, ) is on the line, it is a solution to the equation. Therefore, if we substitute (, ) for and in the equation, we can solve for b. b b b Thus, the slope-intercept form is. Skill Practice 6. Use slope-intercept form to find an equation of a line with slope and passing through,. TIP: We can check the answer to Eample, b graphing the line. Notice that the line appears to pass through (, ) as desired. (, ). The Point-Slope Formula In Eample, we used the slope-intercept form of a line to construct an equation of a line given its slope and a known point on the line. Here we provide another tool called the point-slope formula that (as its name suggests) can accomplish the same result. Suppose a line passes through a given point, and has slope m. If (, ) is an other point on the line, then m m m or m Slope formula Clear fractions. Point-slope formula The Point-Slope Formula The point-slope formula is given b m where m is the slope of the line and, is a known point on the line. Skill Practice Answers 6.

Chapter Linear Equations in Two Variables The point-slope formula is used specificall to find an equation of a line when a point on the line is known and the slope is known. To illustrate the point-slope formula, we will repeat the problem from Eample. Eample Using the Point-Slope Formula to Find an Equation of a Line Use the point-slope formula to find an equation of the line having a slope of and passing through the point (, ). Write the answer in slope-intercept form. Solution: m m and (, ) (, ) Appl the point-slope formula. Simplif. To write the answer in slope-intercept form, clear parentheses and solve for. Clear parentheses. Skill Practice Solve for. The answer is written in slopeintercept form. Notice that this is the same equation as in Eample. 7. Use the point-slope formula to write an equation for a line passing through the point, 6 and with a slope of. Write the answer in slopeintercept form. Eample 6 Finding an Equation of a Line Given Two Points Find an equation of the line passing through the points (, ) and (, ). Write the answer in slope-intercept form. TIP: In Eample 6, the point (, ) was used for (, ) in the point-slope formula. However, either point could have been used. Using the point (, ) for (, ) produces the same final equation: Skill Practice Answers 7. 6 Solution: The slope formula can be used to compute the slope of the line between two points. Once the slope is known, the point-slope formula can be used to find an equation of the line. First find the slope. m Net, appl the point-slope formula. m Substitute m and use either point for (, ). We will use (, ) for (, ). Clear parentheses. Hence, m. Solve for. The final answer is in slope-intercept form.

Section. Equations of a Line Skill Practice 8. Use the point-slope formula to write an equation of the line that passes through the points, and,. Write the answer in slope-intercept form. Eample 7 Finding an Equation of a Line Parallel to Another Line Find an equation of the line passing through the point, to the line 8. Write the answer in slope-intercept form. and parallel Solution: To find an equation of a line, we must know a point on the line and the slope. The known point is,. Because the line is parallel to 8, the two lines must have the same slope. Writing the equation 8 in slopeintercept form, we have 8. Therefore, the slope of both lines must be. We must now find an equation of the line passing through, having a slope of. m Appl the point-slope formula. Substitute m and, for,. 8 Clear parentheses. Write the answer in slope-intercept form. Skill Practice 9. Find an equation of a line containing, and parallel to 7. Write the answer in slope-intercept form. We can verif the answer to Eample 7 b graphing both lines. We see that the line passes through the point, and is parallel to the line 8. See Figure -. 0 8 6 0 8 6 6 8 0 (, ) 8 6 8 0 Figure - Skill Practice Answers 8. 7 9. 9

6 Chapter Linear Equations in Two Variables Eample 8 Finding an Equation of a Line Perpendicular to Another Line Find an equation of the line passing through the point (, ) and perpendicular to the line. Write the answer in slope-intercept form. Solution: The slope of the given line can be found from its slope-intercept form. Solve for. The slope is. The slope of a line perpendicular to this line must be the opposite of the reciprocal of ; hence, m. Using m and the known point (, ), we can appl the point-slope formula to find an equation of the line. m Appl the point-slope formula. Substitute m and (, ) for (, ). 6 Clear parentheses. Solve for. Skill Practice 0. Find an equation of the line passing through the point, 6 and perpendicular to the line 8. Write the answer in slope-intercept form. Calculator Connections From Eample 8, the line should be perpendicular to the line and should pass through the point (, ). Note: In this eample, we are using a square window option, which sets the scale to displa distances on the - and -aes as equal units of measure. Skill Practice Answers 0. 8

Section. Equations of a Line 7. Different Forms of Linear Equations A linear equation can be written in several different forms, as summarized in Table -. Table - Form Eample Comments Standard Form A B C 6 A and B must not both be zero. Horizontal Line k The slope is zero, and the -intercept (k is constant) is (0, k). Vertical Line k The slope is undefined and the -intercept (k is constant) is (k, 0). Slope-Intercept Form Solving a linear equation for results in m b slope-intercept form. The coefficient of Slope is m. Slope the -term is the slope, and the constant -Intercept is (0, b). -Intercept is (0, ). defines the location of the -intercept. Point-Slope Formula m m,, This formula is tpicall used to build Slope is m and, an equation of a line when a point on is a point on the line. the line is known and the slope is known. Although it is important to understand and appl slope-intercept form and the point-slope formula, the are not necessaril applicable to all problems. Eample 9 illustrates how a little ingenuit ma lead to a simple solution. Eample 9 Finding an Equation of a Line Find an equation of the line passing through the point (, ) and perpendicular to the -ais. Solution: An line perpendicular to the -ais must be vertical. Recall that all vertical lines can be written in the form k, where k is constant. A quick sketch can help find the value of the constant (Figure -6). Because the line must pass through a point whose -coordinate is, the equation of the line is. Skill Practice (, ) Figure -6. Write an equation of the line through the point (0, 0) and having a slope of 0. Skill Practice Answers. 0

8 Chapter Linear Equations in Two Variables Section. Boost our GRADE at mathzone.com! Stud Skills Eercises Practice Eercises Practice Problems Self-Tests NetTutor. For this chapter, find the page numbers for the Chapter Review Eercises, the Chapter Test, and the Cumulative Review Eercises. Chapter Review Eercises, page(s) Chapter Test, page(s) Cumulative Review Eercises, page(s) Compare these features and state the advantages of each. e-professors Videos. Define the ke terms. a. Standard form b. Slope-intercept form c. Point-slope formula Review Eercises. Given a. Find the -intercept. b. Find the -intercept. c. Sketch the graph.. Using slopes, how do ou determine whether two lines are parallel?. Using the slopes of two lines, how do ou determine whether the lines are perpendicular? 6. Write the formula to find the slope of a line given two points, and,. 7. Given the two points (, ) and (, ), a. Find the slope of the line containing the two points. b. Find the slope of a line parallel to the line containing the points. c. Find the slope of a line perpendicular to the line containing the points. Concept : Slope-Intercept Form For Eercises 8 7, determine the slope and the -intercept of the line. 8. 9. 7 0. 7 0. 0. 8. 7. 8 9. 9 0 6. 0.6. 7...8

Section. Equations of a Line 9 In Eercises 8, match the equation with the correct graph. 8. 9. 0.... a. b. c. d. e. f. For Eercises, write the equations in slope-intercept form (if possible). Then graph each line, using the slope and -intercept... 6. 6 8 7 6 7 6 7. 8 8. 6 9. 6 6

0 Chapter Linear Equations in Two Variables 0. 0. 0 6 7. Given the standard form of a linear equation A B C, B 0, solve for and write the equation in slope-intercept form. What is the slope of the line? What is the -intercept?. Use the result of Eercise to determine the slope and -intercept of the line 7 9. For Eercises 9, determine if the lines are parallel, perpendicular, or neither... 6 6. 6 0 7..8..6 8. 6 9. 0 9 6 6 For Eercises 0, use the slope-intercept form of a line to find an equation of the line having the given slope and passing through the given point. 0. m,,. m,,. m,,. m,,. m,, 0. m, 0, 0 6. m,, 7. m,, 8. 7 8 m, 0, 9. m, 0, 0. m 0,,. m 0,, Concept : The Point-Slope Formula For Eercises 79, write an equation of the line satisfing the given conditions. Write the answer in slopeintercept form or standard form.. The line passes through the point 0, and has a slope of.. The line passes through the point (0, ) and has a slope of.. The line passes through the point (, 7) and has a slope of.. The line passes through the point (, 0) and has a slope of. 6. The line passes through the point, and has a slope of. 7. The line passes through the point, 6 and has a slope of.

Section. Equations of a Line 8. The line passes through the point 6, and 9. The line passes through the point 7, 7 has a slope of. and has a slope of. 60. The line passes through (0, ) and (, 0). 6. The line passes through (, ) and (, 7). 6. The line passes through (6, ) and (, 0). 6. The line passes through, and 6. The line passes through, and,. 6. The line passes through, and,.,. 66. The line contains the point (, ) and is parallel to a line with a slope of. 67. The line contains the point, and is parallel to a line with a slope of. 68. The line contains the point (, ) and is perpendicular to a line with a slope of. 69. The line contains the point, and is perpendicular to a line with a slope of. 70. The line contains the point, and is parallel to 7. 7. The line contains the point 6, and is parallel to. 7. The line contains the point 8, and is parallel to 8. 7. The line contains the point, and is parallel to 8. 7. The line contains the point (, 0) and is parallel to the line defined b. 7. The line contains the point, 0 and is parallel to the line defined b 6. 76. The line is perpendicular to the line defined b and passes through the point (, ). 77. The line is perpendicular to 7 and passes through the point, 8. 78. The line is perpendicular to and passes through,. 79. The line is perpendicular to and passes through,. Concept : Different Forms of Linear Equations For Eercises 80 87, write an equation of the line satisfing the given conditions. 80. The line passes through, and has 8. The line contains the point, 0 and has an a zero slope. undefined slope. 8. The line contains the point, and has an 8. The line contains the point, 0 and has a undefined slope. zero slope. 8. The line is parallel to the -ais and passes 8. The line is perpendicular to the -ais and passes through (, ). through (, ). 86. The line is parallel to the line and passes 87. The line is parallel to the line and passes through (, ). through,.

Chapter Linear Equations in Two Variables Epanding Your Skills 88. Is the equation in slope-intercept form? Identif the slope and -intercept. 89. Is the equation in slope-intercept form? Identif the slope and -intercept. 90. Is the equation in slope-intercept form? Identif the slope and the -intercept. 9. Is the equation in slope-intercept form? Identif the slope and the -intercept. Graphing Calculator Eercises 9. Use a graphing calculator to graph the lines on 9. Use a graphing calculator to graph the lines on the same viewing window. Then eplain how the the same viewing window. Then eplain how the lines are related. lines are related. 9. Use a graphing calculator to graph the lines on 9. Use a graphing calculator to graph the lines the same viewing window. Then eplain how the on the same viewing window. Then eplain how lines are related. the lines are related. 96. Use a graphing calculator to graph the lines on 97. Use a graphing calculator to graph the lines on a a square viewing window. Then eplain how square viewing window. Then eplain how the lines the lines are related. are related. 98. Use a graphing calculator to graph the equation 99. Use a graphing calculator to graph the equation from Eercise 60. Use an Eval feature to verif from Eercise 6. Use an Eval feature to verif that the line passes through the points (0, ) that the line passes through the points (, ) and (, 0). and (, 7).

Section. Applications of Linear Equations and Graphing Applications of Linear Equations and Graphing. Writing a Linear Model Algebra is a tool used to model events that occur in phsical and biological sciences, sports, medicine, economics, business, and man other fields. The purpose of modeling is to represent a relationship between two or more variables with an algebraic equation. For an equation written in slope-intercept form m b, the term m is the variable term, and the term b is the constant term. The value of the term m changes with the value of (this is wh the slope is called a rate of change). However, the term b remains constant regardless of the value of. With these ideas in mind, a linear equation can be created if the rate of change and the constant are known. Section. Concepts. Writing a Linear Model. Interpreting a Linear Model. Finding a Linear Model from Observed Data Points Eample Finding a Linear Relationship Buffalo, New York, had ft ( in.) of snow on the ground before a snowstorm. During the storm, snow fell at an average rate of 8 in./hr. a. Write a linear equation to compute the total snow depth after hours of the storm. b. Graph the equation. c. Use the equation to compute the depth of snow after 8 hr. d. If the snow depth was. in. at the end of the storm, determine how long the storm lasted. Solution: a. The constant or base amount of snow before the storm began is in. The variable amount is given b 8 in. of snow per hour. If m is replaced b 8 and b is replaced b, we have the linear equation m b 8 b. The equation is in slope-intercept form, and the corresponding graph is shown in Figure -7. Snow Depth (in.) 0 0 0 0 8 hr Snow Depth Versus Time in. 8 0 0 6 8 0 6 8 0 Time (hr) Figure -7

Chapter Linear Equations in Two Variables c. 8 8 8 Substitute 8. 9 in. Solve for. The snow depth was 9 in. after 8 hr. The corresponding ordered pair is (8, 9) and can be confirmed from the graph. d. 8. 8 Substitute.. 8. 8 a 8 b Multipl b 8 to clear fractions. 9 60 Clear parentheses. Solve for. The storm lasted for hr. The corresponding ordered pair is (,.) and can be confirmed from the graph. Skill Practice. When Joe graduated from college, he had $000 in his savings account. When he began working, he decided he would add $0 per month to his savings account. a. Write a linear equation to compute the amount of mone in Joe s account after months of saving. b. Use the equation to compute the amount of mone in Joe s account after 6 months. c. Joe needs $60 for a down pament for a car. How long will it take for Joe s account to reach this amount?. Interpreting a Linear Model Eample Interpreting a Linear Model Skill Practice Answers a. 0 000 b. $70 c. 8 months In 98, President Franklin D. Roosevelt signed a bill enacting the Fair Labor Standards Act of 98 (FLSA). In its final form, the act banned oppressive child labor and set the minimum hourl wage at cents and the maimum workweek at hr. Over the ears, the minimum hourl wage has been increased b the government to meet the rising cost of living. The minimum hourl wage (in dollars per hour) in the United States between 960 and 00 can be approimated b the equation 0.0 0.8 0

Section. Applications of Linear Equations and Graphing Minimum Wage ($/hr) 6 Minimum Wage Earnings per Hour Versus Year 0.0 0.8 0 0 0 0 0 0 0 Year ( 0 corresponds to 960) Figure -8 where represents the number of ears since 960 ( 0 corresponds to 960, corresponds to 96, and so on) (Figure -8). a. Find the slope of the line and interpret the meaning of the slope in the contet of this problem. b. Find the -intercept of the line and interpret the meaning of the -intercept in the contet of this problem. c. Use the linear equation to approimate the minimum wage in 98. d. Use the linear equation to predict the minimum wage in the ear 00. Solution: a. The equation 0.0 0.8 is written in slope-intercept form. The slope is 0.0 and indicates that minimum hourl wage rose an average of $0.0 per ear between 960 and 00. b. The -intercept is (0, 0.8). The -intercept indicates that the minimum wage in the ear 960 0 was approimatel $0.8 per hour. (The actual value of minimum wage in 960 was $.00 per hour.) c. The ear 98 is ears after the ear 960. Substitute into the linear equation. 0.0 0.8 0.0 0.8.0 0.8. Substitute. According to the linear model, the minimum wage in 98 was approimatel $. per hour. (The actual minimum wage in 98 was $. per hour.) d. The ear 00 is 0 ears after the ear 960. Substitute 0 into the linear equation. 0.0 0.8 0.00 0.8.8 Substitute 0. According to the linear model, minimum wage in 00 will be approimatel $.8 per hour provided the linear trend continues. (How does this compare with the current value for minimum wage?)

6 Chapter Linear Equations in Two Variables Skill Practice. The cost of long-distance service with a certain phone compan is given b the equation 0. 6.9, where represents the monthl cost in dollars and represents the number of minutes of long distance. a. Find the slope of the line, and interpret the meaning of the slope in the contet of this problem. b. Find the -intercept and interpret the meaning of the -intercept in the contet of this problem. c. Use the equation to determine the cost of using min of long-distance service in a month.. Finding a Linear Model from Observed Data Points Graphing a set of data points offers a visual method to determine whether the points follow a linear pattern. If a linear trend eists, we sa that there is a linear correlation between the two variables. The better the points line up, the stronger the correlation.* When two variables are correlated, it is often desirable to find a mathematical equation (or model) to describe the relationship between the variables. Eample Writing a Linear Model from Observed Data Figure -9 represents the winning gold medal times for the women s 00-m freestle swimming event for selected summer Olmpics. Let represent the winning time in seconds and let represent the number of ears since 900 ( 0 corresponds to 900, corresponds to 90, and so on). Time (sec) Women's 00-m Freestle for Selected Olmpics 80 70 (7, 8.6) 60 (, 7.) 0 0 0 0 0 0 0 0 0 0 0 0 60 70 80 90 00 0 Year ( 0 corresponds to 900) Figure -9 Skill Practice Answers a. The slope is 0.. This means that the monthl cost increases b cents per minute. b. The -intercept is (0, 6.9). The cost of the long-distance service is $6.9 if 0 min is used. c. $. In 9, the winning time was 7. sec. This corresponds to the ordered pair (, 7.). In 97, the winning time was 8.6 sec, ielding the ordered pair (7, 8.6). a. Use these ordered pairs to find a linear equation to model the winning time versus the ear. b. What is the slope of the line, and what does it mean in the contet of this problem? *The strength of a linear correlation can be measured mathematicall b using techniques often covered in statistics courses.

Section. Applications of Linear Equations and Graphing 7 c. Use the linear equation to approimate the winning time for the 96 Olmpics. d. Would it be practical to use the linear model to predict the winning time in the ear 08? Solution: a. The slope formula can be used to compute the slope of the line between the two points. (Round the slope to decimal places.) m 7. 0.9 7. 0.9 6.96 (, 7.) and (7, 8.6), and, m 8.6 7. 0.87 7 0.9 6.96 7. 0.9 79.6 Hence, m L 0.9. Appl the point-slope formula, using m 0.9 and the point (, 7.). Clear parentheses. Solve for. The answer is in slope-intercept form. b. The slope is 0.9 and indicates that the winning time in the women s 00-m Olmpic freestle event has decreased on average b 0.9 sec/r during this period. c. The ear 96 is 6 ears after the ear 900. Substitute 6 into the linear model. 0.9 79.6 0.96 79.6 8.6 79.6 60.8 Substitute 6. According to the linear model, the winning time in 96 was approimatel 60.8 sec. (The actual winning time in 96 was set b Dawn Fraser from Australia in 9. sec. The linear equation can onl be used to approimate the winning time.) d. It would not be practical to use the linear model 0.9 79.6 to predict the winning time in the ear 08. There is no guarantee that the linear trend will continue beond the last observed data point in 00. In fact, the linear trend cannot continue indefinitel; otherwise, the swimmers times would eventuall be negative. The potential for error increases for predictions made beond the last observed data value. Skill Practice. The figure shows data relating the cost of college tetbooks in dollars to the number of pages in the book. Let represent the cost of the book, and let represent the number of pages. a. Use the ordered pairs indicated in the figure to write a linear equation to model the cost of tetbooks versus the number of pages. b. Cost ($) Cost of Tetbook versus Number of Pages 0 (00, 07) 00 80 (00, 7) 60 0 0 0 0 00 00 00 00 00 Number of Pages Skill Practice Answers a. 0. 7 b. $97

8 Chapter Linear Equations in Two Variables Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. On test da, take a look at an formulas or important points that ou had to memorize before ou enter the classroom. Then when ou sit down to take our test, write these formulas on the test or on scrap paper. This is called a memor dump. Write down the formulas from Chapter. Review Eercises For Eercises, a. Find the slope (if possible) of the line passing through the two points. b. Find an equation of the line passing through the two points. Write the answer in slope-intercept form (if possible) and in standard form. c. Graph the line b using the slope and -intercept. Verif that the line passes through the two given points.., 0 and,., and,., and,., and, 0 Concept : Writing a Linear Model 6. A car rental compan charges a flat fee of $9.9 plus $0.0 per mile. a. Write an equation that epresses the cost of renting a car if the car is driven for miles.

Section. Applications of Linear Equations and Graphing 9 b. Graph the equation. Cost ($) 60 0 0 0 0 0 0 0 60 80 00 0 0 60 80 00 Miles c. What is the -intercept and what does it mean in the contet of this problem? d. Using the equation from part (a), find the cost of driving the rental car 0, 00, and 00 mi. e. Find the total cost of driving the rental car 00 mi if the sales ta is 6%. f. Is it reasonable to use negative values for in the equation? Wh or wh not? 7. Ale is a sales representative and earns a base salar of $000 per month plus a % commission on his sales for the month. a. Write a linear equation that epresses Ale s monthl salar in terms of his sales. b. Graph the equation. Salar ($) 000 00 000 00 000 00 0,000 0,000 Sales ($) 0,000 0,000 0,000 c. What is the -intercept and what does it represent in the contet of this problem? d. What is the slope of the line and what does it represent in the contet of this problem? e. How much will Ale make if his sales for a given month are $0,000? 8. Ava recentl purchased a home in Crescent Beach, Florida. Her propert taes for the first ear are $7. Ava estimates that her taes will increase at a rate of $ per ear. a. Write an equation to compute Ava s earl propert taes. Let be the amount she pas in taes, and let be the time in ears. b. Graph the line. Taes ($) 000 00 000 00 000 00 000 00 6 8 0 6 8 0 Time (Years) c. What is the slope of this line? What does the slope of the line represent in the contet of this problem? d. What is the -intercept? What does the -intercept represent in the contet of this problem? e. What will Ava s earl propert ta be in 0 ears? In ears?

60 Chapter Linear Equations in Two Variables 9. Luigi Luna has started a chain of Italian restaurants called Luna Italiano. He has 9 restaurants in various locations in the northeast United States and Canada. He plans to open five new restaurants per ear. a. Write a linear equation to epress the number of restaurants,, Luigi opens in terms of the time in ears,. b. How man restaurants will he have in ears? c. How man ears will it take him to have 00 restaurants? Concept : Interpreting a Linear Model 0. Sound travels at approimatel one-fifth of a mile per second. Therefore, for ever -sec difference between seeing lightning and hearing thunder, we can estimate that a storm is approimatel mi awa. Let represent the distance (in miles) that a storm is from an observer. Let represent the difference in time between seeing lightning and hearing thunder. Then the distance of the storm can be approimated b the equation 0., where 0. a. Use the linear model provided to determine how far awa a storm is for the following differences in time between seeing lightning and hearing thunder: sec, sec, and 6 sec. b. If a storm is. mi awa, how man seconds will pass between seeing lightning and hearing thunder?. The force (in pounds) required to stretch a particular spring inches beond its rest (or equilibrium ) position is given b the equation., where 0 0. a. Use the equation to determine the amount of force necessar to stretch the spring 6 in. from its rest position. How much force is necessar to stretch the spring twice as far? b. If lb of force is eerted on the spring, how far will the spring be stretched?. The figure represents the median cost of new privatel owned, one-famil houses sold in the midwest from 980 to 00. Price ($000) 0 00 0 00 0 Median Cost of New One-Famil Houses Sold in the Midwest, 980 00. 6. 0 0 0 0 0 Year ( 0 corresponds to 980) Source: U.S. Bureau of the Census and U.S. Department of Housing and Urban Development. Let represent the median cost of a new privatel owned, one-famil house sold in the midwest. Let represent the ear, where 0 corresponds to the ear 980, represents 98, and so on. Then the median cost of new privatel owned, one-famil houses sold in the midwest can be approimated b the equation. 6., where 0. a. Use the linear equation to approimate the median cost of new privatel owned, one-famil houses in the midwest for the ear 00.

Section. Applications of Linear Equations and Graphing 6 b. Use the linear equation to approimate the median cost for the ear 988, and compare it with the actual median cost of $0,600. c. What is the slope of the line and what does it mean in the contet of this problem? d. What is the -intercept and what does it mean in the contet of this problem?. Let represent the average number of miles driven per ear for passenger cars in the United States between 980 and 00. Let represent the ear where 0 corresponds to 980, corresponds to 98, and so on. The average earl mileage for passenger cars can be approimated b the equation 9060, where 0. a. Use the linear equation to approimate the average earl mileage for passenger cars in the United States in the ear 00. b. Use the linear equation to approimate the average mileage for the ear 98, and compare it with the actual value of 9700 mi. c. What is the slope of the line and what does it mean in the contet of this problem? d. What is the -intercept and what does it mean in the contet of this problem? Miles Driven Average Yearl Mileage for Passenger Cars, United States, 980 00,000,000 0,000 8,000 9060 6,000,000,000 0 0 0 0 Year ( 0 corresponds to 980) Concept : Finding a Linear Model from Observed Data Points. The figure represents the winning heights for men s pole vault in selected Olmpic games. Height (m) 8 6 (0,.) Winning Pole Vault Height for Selected Olmpic Games (96,.9) 0 0 0 0 60 80 00 0 Year ( 0 corresponds to 900) a. Let represent the winning height. Let represent the ear, where 0 corresponds to the ear 900, represents 90, and so on. Use the ordered pairs given in the graph (0,.) and (96,.9) to find a linear equation to estimate the winning pole vault height versus the ear. (Round the slope to three decimal places.) b. Use the linear equation from part (a) to approimate the winning vault for the 90 Olmpics. c. Use the linear equation to approimate the winning vault for 976. d. The actual winning vault in 90 was.09 m, and the actual winning vault in 976 was. m. Are our answers from parts (b) and (c) different from these? Wh? e. What is the slope of the line? What does the slope of the line mean in the contet of this problem?

6 Chapter Linear Equations in Two Variables. The figure represents the winning time for the men s 00-m freestle swimming event for selected Olmpic games. Time (sec) Winning Times for Men's 00-m Freestle Swimming for Selected Olmpics 60 0 (0, 7.) 0 0 (8, 8.7) 0 0 0 0 0 0 0 0 0 60 Year ( 0 corresponds to 98) a. Let represent the winning time. Let represent the number of ears since 98 (where 0 corresponds to the ear 98, represents 9, and so on). Use the ordered pairs given in the graph (0, 7.) and (8, 8.7) to find a linear equation to estimate the winning time for the men s 00-m freestle versus the ear. (Round the slope to decimal places.) b. Use the linear equation from part (a) to approimate the winning 00-m time for the ear 97, and compare it with the actual winning time of. sec. c. Use the linear equation to approimate the winning time for the ear 988. d. What is the slope of the line and what does it mean in the contet of this problem? e. Interpret the meaning of the -intercept of this line in the contet of this problem. Eplain wh the men s swimming times will never reach the -intercept. Do ou think this linear trend will continue for the net 0 ears, or will the men s swimming times begin to level off at some time in the future? Eplain our answer. 6. At a high school football game in Miami, hot dogs were sold for $.00 each. At the end of the night, it was determined that 60 hot dogs were sold. The following week, the price of hot dogs was raised to $.0, and this resulted in fewer sales. Onl 7 hot dogs were sold. a. Make a graph with the price of hot dogs on the -ais and the corresponding sales on the -ais. Graph the points (.00, 60) and (.0, 7), using suitable scaling on the - and -aes. Number of Hot Dogs Sold 000 900 800 700 600 00 00 00 00 00 0.0.00.0.00 Price of Hot Dogs ($) b. Find an equation of the line through the given points. Write the equation in slope-intercept form. c. Use the equation from part (b) to predict the number of hot dogs that would sell if the price were changed to $.70 per hot dog. 7. At a high school football game, soft drinks were sold for $0.0 each. At the end of the night, it was determined that 00 drinks were sold. The following week, the price of drinks was raised to $0.7, and this resulted in fewer sales. Onl 80 drinks were sold. a. Make a graph with the price of drinks on the -ais and the corresponding sales per night on the -ais. Graph the points (0.0, 00) and (0.7, 80), using suitable scaling on the - and -aes.

Section. Applications of Linear Equations and Graphing 6 Number of Drinks Sold 600 00 00 000 800 600 00 00 0. 0.0 0.7.00 Price of Drinks ($) b. Find an equation of the line through the given points. Write the equation in slope-intercept form. c. Use the equation from part (b) to predict the number of drinks that would sell if the price were changed to $0.8 per drink. Epanding Your Skills 8. Loraine is enrolled in an algebra class that meets das per week. Her instructor gives a test ever Frida. Loraine has a stud plan and keeps a portfolio with notes, homework, test corrections, and vocabular. She also records the amount of time per da that she studies and does homework. The following data represent the amount of time she studied per da and her weekl test grades. a. Graph the points on a rectangular coordinate sstem. Do the data points appear to follow a linear trend? Time Studied per Da Weekl Test Grade (min) (percent) 60 69 70 7 80 79 90 8 00 89 Test Score (%) 90 80 70 60 0 0 0 0 0 0 0 0 0 0 60 70 80 90 00 Minutes b. Find a linear equation that relates Loraine s weekl test score to the amount of time she studied per da. (Hint: Pick two ordered pairs from the observed data, and find an equation of the line through the points.) c. How man minutes should Loraine stud per da in order to score at least 90% on her weekl eamination? Would the equation used to determine the time Loraine needs to stud to get 90% work for other students? Wh or wh not? d. If Loraine is onl able to spend hr/da studing her math, predict her test score for that week. Points are collinear if the lie on the same line. For Eercises 9, use the slope formula to determine if the points are collinear. 9., 0, 9, 0.,,,. 0,,, 6., 0,,

6 Chapter Linear Equations in Two Variables Graphing Calculator Eercises. Use a Table feature to confirm our answers to. Use a Table feature to confirm our answers to Eercise. Eercise 0(a).. Graph the line 800 0 on the viewing window defined b 0 and 0 600. Use the Trace ke to support our answer to Eercise 7 b showing that the line passes through the points (0.0, 00) and (0.7, 80). 6. Graph the line 0 000 on the viewing window defined b 0 and 0 000. Use the Trace ke to support our answer to Eercise 6 b showing that the line passes through the points (.00, 60) and (.0, 7).

Summar 6 Chapter SUMMARY Section. Ke Concepts Graphical representation of numerical data is often helpful to stud problems in real-world applications. A rectangular coordinate sstem is made up of a horizontal line called the -ais and a vertical line called the -ais. The point where the lines meet is the origin. The four regions of the plane are called quadrants. The point, is an ordered pair. The first element in the ordered pair is the point s horizontal position from the origin. The second element in the ordered pair is the point s vertical position from the origin. The Rectangular Coordinate Sstem and Midpoint Formula Eamples Eample Quadrant II (, ) 6 6 6 Quadrant III 6 -ais Quadrant I Origin Quadrant IV -ais The midpoint between two points is found b using the formula a, b Eample Find the midpoint between (, ) and (, 7). a, 7 b,

66 Chapter Linear Equations in Two Variables Section. Linear Equations in Two Variables Ke Concepts A linear equation in two variables can be written in the form A B C, where A, B, and C are real numbers and A and B are not both zero. The graph of a linear equation in two variables is a line and can be represented in the rectangular coordinate sstem. Eamples Eample Complete a table of ordered pairs. 0 0 9 An -intercept is a point (a, 0) where a graph intersects the -ais. To find an -intercept, substitute 0 for in the equation and solve for. A -intercept is a point (0, b) where a graph intersects the -ais. To find a -intercept, substitute 0 for in the equation and solve for. Eample Given the equation, -intercept: -intercept: 8 0 8 8, 0 0 8 8 8 a0, 8 b A vertical line can be written in the form k. A horizontal line can be written in the form k. Eample Eample

Summar 67 Section. Slope of a Line Ke Concepts The slope of a line m between two distinct points, and, is given b m, 0 The slope of a line ma be positive, negative, zero, or undefined. Eamples Eample The slope of the line between (, ) and (, 7) is m 7 0 Eample Positive slope Negative slope Zero slope Undefined slope Two parallel (nonvertical) lines have the same slope: m m. Two lines are perpendicular if the slope of one line is the opposite of the reciprocal of the slope of the other line: m m or equivalentl, m m. Eample The slopes of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. a. m 7 and m 7 Parallel b. m c. m and m and m Perpendicular Neither

68 Chapter Linear Equations in Two Variables Section. Equations of a Line Ke Concepts Standard Form: A B C (A and B are not both zero) Horizontal line: k Vertical line: k Slope-intercept form: m b Point-slope formula: m Slope-intercept form is used to identif the slope and -intercept of a line when the equation is given. Slope-intercept form can also be used to graph a line. Eamples Eample Find the slope and -intercept. Then graph the equation. 7 7 The slope is 7 Up 7 Solve for. Solve-intercept form 7 ; the -intercept is 0,. Right (0, ) Start here The point-slope formula can be used to construct an equation of a line, given a point and a slope. Eample Find an equation of the line passing through the point (, ) and having slope m. Using the point-slope formula gives m 8

Summar 69 Section. Ke Concepts A linear model can often be constructed to describe data for a given situation. Applications of Linear Equations and Graphing Eamples Eample The graph shows the average per capita income in the United States for 980 00. The ear 980 corresponds to 0 and income is measured in dollars. Dollars ($) Average per Capita Yearl Income in United States (980 00),000 0,000,000 (0,,6) 0,000,000 0 0 (,,0) 0 0 Year ( 0 corresponds to 980) Given two points from the data, use the point-slope formula to find an equation of the line. Interpret the meaning of the slope and -intercept in the contet of the problem. Write an equation of the line, using the points (,,0) and (0,,6). Slope:,6,0 0, 76,0 76,0 76 780 76 7 The slope m 76 indicates that the average income has increased b $76 per ear. The -intercept (0, 7) means that the average income in 980 0 was $7. Use the equation to predict values. Predict the average income for 00 0. 760 7 9,9 According to this model, the average income in 00 will be approimatel $9,9.

70 Chapter Linear Equations in Two Variables Chapter Review Eercises Section. 6. 0. Label the following on the diagram: a. Origin b. -Ais c. -Ais d. Quadrant I e. Quadrant II f. Quadrant III g. Quadrant IV 0 7. 6 7 6. Find the midpoint of the line segment between the two points (, ) and (, 8).. Find the midpoint of the line segment between the two points (.,.7) and (., 8.).. Determine the coordinates of the points labeled in the graph. 0 Section. For Eercises 7, complete the table and graph the line defined b the points.. 6 F B E A D C G For Eercises 8, graph the lines. In each case find at least three points and identif the - and -intercepts (if possible). 8. 6 0 0 9. 0

Review Eercises 7 0. 6 c. 6. Draw a line with slope (answers ma var).. 6. Draw a line with slope (answers ma var). Section.. Find the slope of the line. a. b. For Eercises 8, find the slope of the line that passes through each pair of points.., 6,, 0 6. 7,,, 7. 8,,, 8. a, b,, 9. Two points for each of two lines are given. Determine if the lines are parallel, perpendicular, or neither. L :, 6 and, L :, and (7, 0)

7 Chapter Linear Equations in Two Variables For Eercises 0, the slopes of two lines are given. Based on the slopes, are the lines parallel, perpendicular, or neither? 0. m. m, m, m. m 7, m 7. The graph indicates that the enrollment for a small college has been increasing linearl between 990 and 00. a. Use the two data points to find the slope of the line. b. Interpret the meaning of the slope in the contet of this problem. Number of Students. Find the slope of the stairwa pictured here. Section. College Enrollment b Year 000 00 (00, 8) 000 00 (990, 00) 000 00 0 98 990 99 000 00 00. Write an equation for each of the following. a. Horizontal line b. Point-slope formula c. Standard form d. Vertical line 8 in. e. Slope-intercept form 6 in. For Eercises 6 0, write our answer in slopeintercept form or in standard form. 6. Write an equation of the line that has slope and -intercept (0, 6). 7. Write an equation of the line that has slope and -intercept (, 0). 8. Write an equation of the line that passes through the points ( 8, ) and (, 9). 9. Write an equation of the line that passes through the point (6, ) and is perpendicular to the line. 0. Write an equation of the line that passes through the point (0, ) and is parallel to the line.. For each of the given conditions, find an equation of the line a. Passing through the point (, ) and parallel to the -ais. b. Passing through the point (, ) and parallel to the -ais. c. Passing through the point (, ) and having an undefined slope. d. Passing through the point (, ) and having a zero slope.. Are an of the lines in Eercise the same? Section.. Keosha loves the beach and decides to spend the summer selling various ice cream products on the beach. From her accounting course, she knows that her total cost is calculated as Total cost fied cost variable cost She estimates that her fied cost for the summer season is $0 per da. She also knows that each ice cream product costs her $0. from her distributor. 9

Test 7 a. Write a relationship for the dail cost in terms of the number of ice cream products sold per da. b. Graph the equation from part (a) b letting the horizontal ais represent the number of ice cream products sold per da and letting the vertical ais represent the dail cost. Cost ($) 00 80 60 0 0 00 80 60 0 0 0 00 0 00 0 00 0 00 0 00 Number of Ice Cream Products c. What does the -intercept represent in the contet of this problem? d. What is her cost if she sells 0 ice cream products? e. What is the slope of the line? f. What does the slope of the line represent in the contet of this problem?. The margin of victor for a certain college football team seems to be linearl related to the number of rushing ards gained b the star running back. The table shows the statistics. Margin of Victor Yards Rushed Margin of Victor 00 0 60 0 0 0 7 a. Graph the data to determine if a linear trend eists. Let represent the number of ards rushed b the star running back and represent the points in the margin of victor. 0 0 60 80 00 0 Yards Rushed b. Find an equation for the line through the points (0, 7) and (00, 0). c. Based on the equation, what would be the result of the football game if the star running back did not pla? Chapter Test. Given the equation 6, complete the ordered pairs and graph the corresponding points. 0,, 0, 6 7 8 9 6 7 8. Determine whether the following statements are true or false and eplain our answer. a. The product of the - and -coordinates is positive onl for points in quadrant I. b. The quotient of the - and -coordinates is negative onl for points in quadrant IV. c. The point (, ) is in quadrant III. d. The point (0, 0) lies on the -ais.. Find the midpoint of the line segment between the points (, ) and (, ).

7 Chapter Linear Equations in Two Variables. Eplain the process for finding the - and -intercepts. For Eercises 8, identif the - and -intercepts (if possible) and graph the line.. 6. 6 8 7 6 0. Describe the relationship of the slopes of a. Two parallel lines b. Two perpendicular lines. The slope of a line is 7. a. Find the slope of a line parallel to the given line. b. Find the slope of a line perpendicular to the given line.. Two points are given for each of two lines. Determine if the lines are parallel, perpendicular, or neither. L :, and, 6 L :, 0 and (0, ). Given the equation, a. Write the line in slope-intercept form. b. Determine the slope and -intercept. c. Graph the line, using the slope and -intercept. 7. 8. 6 6 9. Find the slope of the line, given the following information: a. The line passes through the points (7, ) and (, 8). b. The line is given b 6.. Determine if the lines are parallel, perpendicular, or neither. a. b. 9 0 c. 6 d. 9 0. 0. Write an equation that represents a line subject to the following conditions. (Answers ma var.) a. A line that does not pass through the origin and has a positive slope b. A line with an undefined slope

Cumulative Review Eercises 7 c. A line perpendicular to the -ais. What is the slope of such a line? d. A slanted line that passes through the origin and has a negative slope 6. Write an equation of the line that passes through the point 8, with slope. Write the answer in slope-intercept form. 7. Write an equation of the line containing the points, and (, 0). 8. Write an equation of a line containing, and parallel to 6. 9. Write an equation of the line that passes through the point ( 0, ) and is perpendicular to 7. Write the answer in slope-intercept form. 0. Jack sells used cars. He is paid $800 per month plus $00 commission for each automobile he sells. a. Write an equation that represents Jack s monthl earnings in terms of the number of automobiles he sells. b. Graph the linear equation ou found in part (a). Monthl Earnings ($) 600 600 800 000 00 00 600 800 6 8 0 6 8 0 Number of Cars c. What does the -intercept mean in the contet of this problem? d. How much will Jack earn in a month if he sells 7 automobiles?. The following graph represents the life epectanc for females in the United States born from 90 through 00. Life Epectanc (ears) 90 80 70 60 Life Epectanc for Females in the United States According to Birth Year (0, 66) (0, 7) 0 0 0 0 0 0 0 60 70 Year of Birth ( 0 corresponds to 90) Source: National Center for Health Statistics a. Determine the -intercept from the graph. What does the -intercept represent in the contet of this problem? b. Using the two points (0, 66) and (0, 7), determine the slope of the line. What does the slope of the line represent in the contet of this problem? c. Use the -intercept and the slope found in parts (a) and (b) to write an equation of the line b letting represent the ear of birth and represent the corresponding life epectanc. d. Using the linear equation from part (c), approimate the life epectanc for women born in the United States in 99. How does our answer compare with the reported life epectanc of 79 ears? Chapters Cumulative Review Eercises. Simplif the epression. 7. Simplif the epression 8 9 6.. Solve the equation for z. z z z. Solve the equation for b. b 6 b. A bike rider pedals 0 mph to the top of a mountain and mph on the wa down. The total time for the round trip is 0 hr. Find the distance to the top of the hill.

76 Chapter Linear Equations in Two Variables 6. The formula for the volume of a right circular clinder is V pr h. a. Solve for h. b. Find h if a soda can contains cm (which is approimatel oz) of soda and the diameter is 6.6 cm. Round the answer to decimal place. 7. Solve the inequalities. Write our answers in interval notation. 8. Find the slope of the line that passes through the points, and 6,. 9. Find the midpoint of the line segment with endpoints, and 0,. For Eercises 0, a. find the - and -intercept, b. find the slope, and c. graph the line. 0. a. b. 7 0. 0. Find an equation for the vertical line that passes through the point (7, ).. Find an equation for the horizontal line that passes through the point (9, 0).. Find an equation of the line passing through (, ) and parallel to 6. Write the answer in slope-intercept form.. Find an equation of the line passing through (, ) and perpendicular to. Write the answer in slope-intercept form. 6. At the movies, Laquita paid for drinks and popcorn for herself and her two children. She spent twice as much on popcorn as on drinks. If her total bill came to $7.9, how much did she spend on drinks and how much did she spend on popcorn? 7. Three raffle tickets are represented b three consecutive integers. If the sum of the ticket numbers is 776, find the three numbers. 8. A chemist mies a 0% salt solution with a 0% salt solution to get L of a 8% salt solution. How much of the 0% solution and how much of the 0% solution did she use? 9. The earl rainfall for Seattle, Washington, is 0.7 in. less than twice the earl rainfall for Los Angeles, California. If the total earl rainfall for the two cities is 0 in., how much rain does each cit get per ear? 0. Simplif. 76