Linear Equations in Two Variables


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1 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 datoda 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
2 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 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 Quadrant III Quadrant IV 6 Figure  ais
3 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 (, ) (, )
4 06 Chapter Linear Equations in Two Variables Savings ($) 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 (, ) (, )
5 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, 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 Ranger (, ) Station Hiker 6 Figure  Skill Practice Answers. a. 0.7,., b
6 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 SelfTests 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 lefthand side of the paper. Then give a written eplanation on the righthand 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. eprofessors Videos Perform the correct math here. () 0 Eplain the process here. Do multiplication before addition.
7 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 A B C E D E A C B D
8 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 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 Number of Cases of West Nile Virus (Colorado, 00) 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?
9 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, ) (, ) (, ) (, ) (, ) (, ) 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, ,. and., 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
10 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
11 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 (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
12 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 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
13 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 righthand 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 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, )
14 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 To find the intercept, substitute 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.
15 Section. Linear Equations in Two Variables 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 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)
16 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 straightline 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 $,000 per ear. The equation that describes the depreciation line is where represents the value of the truck in dollars and is the age of the truck in ears. a. Find the  and intercepts. Plot the intercepts on a rectangular coordinate sstem, and draw the line that represents the straightline depreciation. b. What does the intercept represent in the contet of this problem? c. What does the intercept represent in the contet of this problem? Solution: a. To find the intercept, substitute To find the intercept, substitute ,000,000,000 6,000 6,000,000 6,000,0000 6,000 The intercept is (, 0). The intercept is (0, 6,000). Taable Value ($) Taable Value of a Truck Versus the Age of the Vehicle 70,000 60,000 0,000 0,000 0,000 0,000 0, 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.
17 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 ) 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.
18 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 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
19 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 SelfTests NetTutor eprofessors 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
20 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 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
21 Section. Linear Equations in Two Variables In Eercises 8, graph the linear equation
22 Chapter Linear Equations in Two Variables 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
23 Section. Linear Equations in Two Variables A salesperson makes a base salar of $0,000 a ear plus a % commission on the total sales for the ear. The earl salar can be epressed as a linear equation as 0, where represents the earl salar and represents the total earl sales. Total Yearl Salar ($) 60,000 0,000 0,000 Yearl Salar Relating to Sales ,000 00, , ,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?
24 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 Number of Miles Concept : Horizontal and Vertical Lines For Eercises, identif the line as either vertical or horizontal, and graph the line
25 Section. Linear Equations in Two Variables 7 Epanding Your Skills For Eercises, find the  and intercepts 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 For Eercises 60 6, use a graphing calculator to graph the lines on the suggested viewing window 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..
26 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. 8ft change in Figure 6 ft change in which is greater than the slope of the ramp, which
27 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.
28 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, ) (, ) 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
29 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.
30 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
31 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.
32 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.) (970,.0) 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
33 Section. Slope of a Line Solution: a. 970,.0 and 00, 6.,, m Label the points. Appl the slope formula. m. or m 0.6 b. The slope is approimatel 0.6, meaning that the fulltime 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 ( Name two features that this online service offers that can help ou in this course.. Define the ke term slope. Practice Eercises Practice Problems SelfTests NetTutor eprofessors 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
34 6 Chapter Linear Equations in Two Variables 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. and 0., 0.. (, ) and (, 7)
35 Section. Slope of a Line 7., and, 0., and,. 8, and (, ) 6..6,. and (0, 6.) 7. (., ) and., 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 Concept : Parallel and Perpendicular Lines 8. Can the slopes of two perpendicular lines both be positive? Eplain our answer.
36 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 Number of Cellular Phone Subscriptions (998, 70) (006, 0) Number of SUVs Sold in the United States (99, ) (000, )
37 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) Average Weight for Bos b Age (,.) Age (r) (0, 7.) 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.) (,.) 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
38 0 Chapter Linear Equations in Two Variables Section. Concepts. SlopeIntercept Form. The PointSlope Formula. Different Forms of Linear Equations Equations of a Line. SlopeIntercept 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 slopeintercept 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 : slopeintercept form. SlopeIntercept Form of a Line m b is the slopeintercept form of a line. m is the slope and the point (0, b) is the intercept. The equation 7 is written in slopeintercept 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 slopeintercept form, then find the slope and intercept. Solution: Write the equation in slopeintercept 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 slopeintercept form. Determine the slope and the intercept.
39 Section. Equations of a Line The slopeintercept 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 slopeintercept 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.
40 Chapter Linear Equations in Two Variables Solution: a. The equations are written in slopeintercept 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 slopeintercept 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 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 Add to both sides. Skill Practice Given the pair of equations, determine if the lines are parallel, perpendicular, or neither Eample Using SlopeIntercept Form to Find an Equation of a Line Use slopeintercept 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 slopeintercept 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 slopeintercept form becomes m b b
41 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 slopeintercept form is. Skill Practice 6. Use slopeintercept 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 PointSlope Formula In Eample, we used the slopeintercept 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 pointslope 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. Pointslope formula The PointSlope Formula The pointslope 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.
42 Chapter Linear Equations in Two Variables The pointslope 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 pointslope formula, we will repeat the problem from Eample. Eample Using the PointSlope Formula to Find an Equation of a Line Use the pointslope formula to find an equation of the line having a slope of and passing through the point (, ). Write the answer in slopeintercept form. Solution: m m and (, ) (, ) Appl the pointslope formula. Simplif. To write the answer in slopeintercept 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 pointslope 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 slopeintercept form. TIP: In Eample 6, the point (, ) was used for (, ) in the pointslope 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 pointslope formula can be used to find an equation of the line. First find the slope. m Net, appl the pointslope formula. m Substitute m and use either point for (, ). We will use (, ) for (, ). Clear parentheses. Hence, m. Solve for. The final answer is in slopeintercept form.
43 Section. Equations of a Line Skill Practice 8. Use the pointslope formula to write an equation of the line that passes through the points, and,. Write the answer in slopeintercept 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 slopeintercept 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 pointslope formula. Substitute m and, for,. 8 Clear parentheses. Write the answer in slopeintercept form. Skill Practice 9. Find an equation of a line containing, and parallel to 7. Write the answer in slopeintercept 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 (, ) Figure  Skill Practice Answers
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