Anytime plan TalkMore plan


 Emmeline Harrison
 1 years ago
 Views:
Transcription
1 CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations is a set of two or more equations with the same variables. A solution of a sstem of equations is a set of values that makes all the equations true. Read the eample in our book and then read the eample below. EXAMPLE The Antime longdistance plan charges $.8 per month plus 5 a minute. The TalkMore plan charges 9 a minute and no monthl fee. For what number of minutes are the charges for the two plans the same? a. Write a sstem of two equations to model this situation. b. Solve the sstem b creating a table. Eplain the realworld meaning of the solution, and locate the solution on a graph. Solution a. Let represent the number of minutes, and let represent the charge in dollars. The charge is the monthl fee plus the rate times the number of minutes. Here is the sstem of equations. b. Create a table from the equations. Fill in the times and calculate the charge for each plan. The table shows that when, both values are.8. Because (,.8) satisfies both equations, it is the solution of the sstem. The solution means that both plans charge $.8 for minutes of longdistance calls. On the graph, the solution is the point where the two lines intersect. Charge (dollars) Antime (,.8) TalkMore (, ) Antime plan TalkMore plan (, 18) 8 16 Time (minutes) (, 1.8) LongDistance Plans Time Antime TalkMore (min) (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 75
2 Previous Net Lesson 6.1 Solving Sstems of Equations (continued) Investigation: Where Will The Meet? Steps 1 In this investigation, two students walk along a 6meter segment. Walker A starts at the.5meter mark and walks toward the 6meter mark at a rate of 1 m/sec. Walker B starts at the meter mark and walks toward the 6meter mark at a rate of.5 m/sec. Here is a graph of the data collected b one group. Steps 5 7 You can model this situation with a sstem of equations and then solve the sstem to figure out when and where Walker A passes Walker B. If represents the time in seconds and represents the distance from the meter mark, the sstem is.5.5 Here are graphs of the equations on the same aes. The graphs intersect at (,.5), indicating that Walker A passes Walker B after seconds, when both walkers are at the.5meter mark. Steps 8 If Walker A moved faster than 1 m/sec, the slope of Walker A s line would increase and the intersection point would move closer to the origin, indicating that Walker A passes Walker B sooner and closer to the meter mark. If the two walkers moved at the same speed, the would never meet. The slope of the lines would be equal, so the lines would be parallel. The sstem of equations for this situation has no solution. 1 Walker A Walker B 1 Distance (m) Distance (m) A A A B B A B A B B A Time (sec) Walker A Walker B Time (sec) Distance (m) 8 6 Walker A Walker B Distance (m) 8 6 Walker B Walker A 6 8 Time (sec) Time (sec) 1 If both walkers walked at the same speed from the same starting mark, the two lines would be identical. Ever point on the line is a solution of the sstem, indicating that the walkers are alwas at the same location at the same time. The investigation shows that two lines can intersect at zero points, at one point, or at ever point. So a sstem of linear equations can have zero, one, or an infinite number of solutions. 76 Discovering Algebra Condensed Lessons Ke Curriculum Press
3 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Substitution Previous Net In this lesson ou will represent situations with sstems of equations use the substitution method to solve sstems of linear equations When ou use a graph or a table to solve a sstem of equations, ou ma onl be able to find an approimate solution. The substitution method allows ou to find an eact solution of a sstem. Read Eample A in our book, which shows how to solve a sstem using the substitution method. Investigation: All Tied Up Start with a thin rope and a thick rope, each 1 meter long. If ou tie knots in each rope, measuring the length after each knot, ou might get data like this. Use the techniques ou learned in Chapter 5 to write a linear equation to model the data for each rope. A possible model for the thin rope is 6, where is the number of knots and is the length in centimeters. The intercept,, is the length of the rope before ou tie an knots. The slope, 6, is the change in the length for each knot. A possible model for the thick rope is.. This equation indicates that the initial length is cm and that the length decreases b. cm for each knot. Now, suppose the initial length of the thin rope is 9 meters and the initial length of the thick rope is meters. This sstem of equations models this situation To estimate the solution of this sstem, make a graph and estimate the point of intersection. The intersection point is about (, 76). You can also find the solution b using the substitution method. Substitute 9 6 (from the first equation) for in the second equation, and solve the resulting equation.. Length of thin rope Length of thick rope Original second equation Substitute 9 6 for. Number of knots 9. Add 6 to both sides and simplif. Thin Rope Length (cm) Length of ropes (cm) 8 6 Number of knots Thick Rope Length (cm) m thin rope m thick rope Number of knots. Subtract from both sides..6 Divide both sides b.. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 77
4 Previous Net Lesson 6. Solving Sstems of Equations Using Substitution (continued) Because represents the number of knots, the solution must be a whole number. So round to. When is, is about 76. So the solution is (, 76). This means that when knots have been tied in each rope, the ropes are about the same length, 76 cm. Think about how the models would be different if the two ropes had the same thickness. In this situation the slopes would be the same, so the lines would be parallel. In this case the sstem would have no solutions. In other words, the ropes would never be the same length. If the ropes had the same thickness and the same starting length, the equations and the lines would be eactl the same. In this case there are man solutions. The ropes would be the same length after an number of knots had been tied. When ou solve a sstem using the substitution method, ou sometimes need to rewrite one of the equations before ou can substitute. Eample B in our book shows ou how to solve a sstem when both equations are given in standard form. Read this eample and the tet that follows carefull. Then, read the eample below. EXAMPLE Use the substitution method to solve this sstem. s 5t s 6 t 5 Solution Rewrite one of the equations so that a variable is alone on one side. s 6 t 5 s 11 t Original second equation. Subtract 5 from both sides. Now, substitute s 11 for t in the first equation and solve for s. s 5(s 11) Substitute s 11 for t. s s 55 Distribute the 5. s s 5 Subtract from 55. 1s 5 Add s to both sides. s Divide both sides b 1. To find the value of t, substitute for s in either equation and solve for t. You should find that the solution of the sstem is (s, t) (, ). Check this solution b substituting it into both equations. 78 Discovering Algebra Condensed Lessons Ke Curriculum Press
5 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Elimination Previous Net In this lesson ou will represent situations with sstems of equations use the elimination method to solve sstems of linear equations Read the tet at the beginning of Lesson 6. in our book. It eplains that ou can add two equations to get another true equation. Then, read Eample A carefull and make sure ou understand it. In the eample, the variable s is eliminated just b adding the equations. As ou will see in the investigation, sometimes using the elimination method requires a bit more work. Investigation: Paper Clips and Pennies Place one paper clip along the long side of a piece of paper. Then, line up enough pennies to complete the 11inch length. If ou use a jumbo paper clip, ou should find that ou need 1 pennies. Place two paper clips along the short side of the sheet of paper, and add pennies to complete the 8.5inch length. With jumbo paper clips, ou ll need 6 pennies. If C is the length of a paper clip and P is the diameter of a penn, ou can write this sstem of equations to represent this situation. C 1P 11 C 6P 8.5 Notice that ou can t eliminate a variable b adding the two original equations. However, look what happens when ou multipl both sides of the first equation b. C 1P 11 C P C 6P 8.5 C 6P 8.5 Because ou multiplied both sides of the first equation b the same number, the new equation has the same solutions as the original. You can now eliminate the variable C b adding the two equations in the new sstem. C P C 6P 8.5 Long side Short side 18P 1.5 Add the equations. P.75 Divide b 18. To find the value of C, substitute.75 for P in either equation and solve for C. C 1(.75) 11 or C 6(.75) 8.5 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 79
6 Previous Net Lesson 6. Solving Sstems of Equations Using Elimination (continued) You should find that C is. Be sure to check the solution b substituting.75 for P and for C in both equations. 1(.75) 11 and () 6(.75) 8.5 The solution (.75, ) means that the penn has a diameter of.75 inch and the paper clip has a length of inches. There are several was ou could have solved the original sstem of equations. For eample, instead of multipling the first equation b, ou could have multiplied the second equation b. Then, the coefficient of P would be 1 in both equations and ou could eliminate P b adding the equations. Read the rest of the lesson in our book. Here is an additional eample. EXAMPLE At Marli s Discount Music Mart, all CDs are the same price and all cassette tapes are the same price. Rashid bought si CDs and five cassette tapes for $ Quinc bought four CDs and nine cassette tapes for $1.7. Write and solve a sstem of equations to find the price of a CD and the price of a cassette tape. Solution If c is the price of a CD and t is the price of a tape, then the problem can be modeled with this sstem. 6c 5t c 9t 1.7 Rashid s purchase Quinc s purchase If ou multipl the first equation b and the second equation b, ou will be able to add the equations to eliminate c. 6c 5t c t 5.56 Multipl both sides b. c 9t 1.7 1c 7t t t 7.98 Multipl both sides b. Add the equations. Divide. To find the value of c, substitute 7.98 for t in either equation and solve for t. 6c 5t Original first equation. 6c 5(7.98) Substitute 7.98 for t. 6c c Multipl. Subtract 9.9 from both sides. c 1.98 Divide both sides b 6. Cassette tapes cost $7.98 and CDs cost $1.98. Be sure to check this solution b substituting it into both original equations. 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
7 CONDENSED L E S S O N 6. Solving Sstems of Equations Using Matrices Previous Net In this lesson ou will represent situations with sstems of equations use matrices to solve sstems of linear equations You now know how to solve sstems of equations with tables and graphs and b using the substitution and elimination methods. You can also solve sstems of equations b using matrices. Pages 1 of our book eplain how to represent a sstem of equations with a matri and then use row operations to find the solution. Read this tet and Eample A carefull. Investigation: Diagonalization Consider this sstem of equations. Because the equations are in standard form, ou can represent the sstem with a matri. Write the numerals from the first equation in the first row, and write the numerals from the second equation in the second row. To solve the equation, perform row operations to get 1 s in the diagonal of the matri and s above and below the diagonal as shown here. To get a as the first entr in the second row, add times the first row to the second row. This step is similar to using the elimination method to eliminate from the second equation. times row 1 6 New matri row New row 8 To get 1 as the second entr in the second row, divide that row b a 1 b From the second row, ou can see that. Now, subtract the second row from the first to get a as the second entr in the first row. This is similar to substituting for in the first equation to get 8. Row New matri row 1 8 New row (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 81
8 Previous Net Lesson 6. Solving Sstems of Equations Using Matrices (continued) To get a 1 as the first entr in the first row, divide the row b. 1 1 You can now see that and. You can check this solution b substituting it into the original equation. Eample B in our book shows that matrices are useful for solving sstems of equations involving large numbers. Here is another eample. EXAMPLE At a college football game, students paid $1 per ticket and nonstudents paid $18 per ticket. The number of students who attended was 1, more than the number of nonstudents. The total of all ticket sales was $67,6. How man of the attendees were students, and how man were nonstudents? Solution If S is the number of students and N is the number of nonstudents, then ou can represent the situation with this sstem and matri. S N 1, 1 1 1, 1S 18N 67, ,6 Use row operations to find the solution. Add 1 times row 1 to row to get 1 1 1, new row. 5, 1 1 1, Divide row b. 1 1,67 1, Add row to row 1 to get new row ,67 The final matri shows that S, and N 1,67. So, students and 1,67 nonstudents attended the game. You can check this solution b substituting it into both original equations. 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
9 CONDENSED L E S S O N 6.5 Inequalities in One Variable Previous Net In this lesson ou will write inequalities to represent situations learn how appling operations to both sides of an inequalit affects the direction of the inequalit smbol solve a problem b writing and solving an inequalit An inequalit is a statement that one quantit is less than or greater than another. Inequalities are written using the smbols,,, and. Read the tet on page 9 of our book, which gives several eamples from everda life and how to write them as inequalities. Just as with equations, ou can solve inequalities b appling the same operations to both sides. However, as ou will learn in the investigation, ou need to be careful about the direction of the inequalit smbol. Investigation: Toe the Line In this investigation, two walkers stand on a number line. Walker A starts on the number, and Walker B starts on the number. You can represent this situation with the inequalit. Steps 1 When an announcer calls out an operation, the walkers perform the operation on their numbers and move to new positions based on the result. The new positions are represented b an inequalit, with the position of Walker A on the left side and the position of Walker B on the right side. The drawings below show the walkers positions after the first two operations along with the corresponding inequalit. Operation: Add ; Inequalit: 6 Operation: Subtract ; Inequalit: 1 A B A B A B This table shows the results of the remaining operations. Walker A s Inequalit Walker B s Operation position smbol position Add 1 1 Subtract 5 Multipl b 6 Subtract 7 1 Multipl b 9 Add 5 8 Divide b 1 Subtract 1 Multipl b 1 1 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 8
10 Previous Net Lesson 6.5 Inequalities in One Variable (continued) Steps 5 9 Notice that when a number is added to or subtracted from the walkers positions, the direction of the inequalit (that is, the relative positions of the walkers) remains the same. The direction of an inequalit also stas the same when the positions are multiplied or divided b a positive number. However, when the positions are multiplied or divided b a negative number, the direction of the inequalit (that is, the relative positions of the walkers) is reversed. Check these findings b starting with another inequalit and appling operations to both sides. You should find that the direction of the inequalit smbol is reversed onl when ou multipl or divide b a negative number. Read Eample A in our book, which shows how to graph solutions to inequalities on a number line. Then, read Eample B, which applies what ou learned in the investigation to solve an inequalit. Here is an additional eample. EXAMPLE A Jack takes the bus to the bowling alle. He has $15 when he arrives. It costs $.5 to bowl one game. If Jack needs $1.5 to take the bus home, how man games can he bowl? Solve this problem b writing and solving an inequalit. Solution Let g represent the number of games Jack can bowl. We know that the amount Jack starts with minus the amount he spends bowling must be at least (that is, greater than or equal to) $1.5. So we can write this inequalit. Amount Jack starts with Cost of bowling g games Bus fare 15.5g 1.5 Now, solve the inequalit. 15.5g 1.5 Original inequalit g Subtract 15 from both sides..5g g g 6 Subtract. Divide both sides b.5, and reverse the inequalit smbol. Divide. Jack can bowl 6 games or fewer. Here, g 6 is graphed on a number line Discovering Algebra Condensed Lessons Ke Curriculum Press
11 CONDENSED L E S S O N 6.6 Graphing Inequalities in Two Variables Previous Net In this lesson ou will graph linear inequalities in two variables You know how to graph linear equations in two variables, such as 6. In this lesson ou will learn to graph linear inequalities in two variables, such as 6 and 6. Investigation: Graphing Inequalities To complete this investigation, ou ll need a grid worksheet like the one on page 7 of our book. Choose one of the statements listed on page 7. For each point shown with a circle on the worksheet, substitute the coordinates of the point into the statement, and then fill in the circle with the relational smbol,,, or, that makes the statement true. For eample, if ou choose the statement 1, do the following for the point (, ): 1 Original statement. 1 () Substitute for and for. 7 Subtract. Because the smbol makes this statement true, write in the circle corresponding to the point (, ). Here are completed grids for the four statements (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 85
12 Previous Net Lesson 6.6 Graphing Inequalities in Two Variables (continued) Notice that for each statement, the circles containing equal signs form a straight line. All the circles above the line are filled in with smbols, and all the circles below the line are filled in with smbols. Choose one of the statements and test a point with fractional or decimal coordinates. For eample, in the grid for 1, (., 1.5) is below the line of equal signs. Substitute the coordinates into the statement. 1 Original statement (.) Substitute. for and 1.5 for Subtract Insert the appropriate smbol. The resulting statement gets a smbol, just like the other points below the line of equal signs. Shown here are graphs of 1, 1, 1, 1, and 1. In each graph the shaded regions include the points that make the statement true. A dashed line indicates that the line is not included in the graph. A solid line indicates that the line is included Make similar graphs for the other inequalities. You should notice the following: Graphs of inequalities in the form epression and epression are shaded above the line. Graphs of inequalities in the form epression and epression are shaded below the line. Graphs of inequalities in the form epression and epression require a solid line. Graphs of inequalities in the form epression and epression require a dashed line. Read the rest of the lesson and the eample in our book. When ou are finished, ou should be able to graph an linear inequalit. 86 Discovering Algebra Condensed Lessons Ke Curriculum Press
13 CONDENSED L E S S O N 6.7 Sstems of Inequalities Previous Net In this lesson ou will graph solutions of sstems of inequalities use sstems of inequalities to represent situations involving constraints You can find the solution of a sstem of equations b graphing the equations and locating the points of intersection. You can use a similar method to find the solution of a sstem of inequalities. Read Eample A in our book. Then, read the additional eample below. EXAMPLE Graph this sstem of inequalities and indicate the solution. Solution Graph with a solid line because its points satisf the inequalit. Shade above the line because its inequalit has the greater than or equal to smbol. Graph with a dashed line because its points do not satisf the inequalit. Shade below the line because in the inequalit is less than the epression in. The points in the overlapping region satisf both inequalities, so the overlapping region is the solution of the sstem. _ 6 Eample B in our book shows how sstems of inequalities are useful for modeling situations involving constraints. Read through the eample. Investigation: A Tpical Envelope Here are two constraints the U.S. Postal Service imposes on envelope sizes. The ratio of length to width must be less than or equal to.5. The ratio of length to width must be greater than or equal to 1.. If l and w represent the length and width of an envelope, then the first constraint l can be represented b the equation w.5 and the second can be represented l b w 1.. You can solve each inequalit for l b multipling both sides b w. This gives the sstem l.5w l 1.w (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 87
14 Previous Net Lesson 6.7 Sstems of Inequalities (continued) Note that ou do not need to reverse the direction of the inequalit smbol when ou multipl both sides b because the width of an envelope must be a positive number. Here, both inequalities are graphed on the same aes. The overlap of the shaded regions is the solution of the sstem. You can check this b choosing a point from the overlapping region and making sure its coordinates satisf both inequalities. Length (in.) 5 l Step 5 in our book gives the dimensions of four envelopes. Points corresponding to these envelopes are plotted on the graph here. Point a, which corresponds to a 5 in.b8 in. envelope, and Point d, which corresponds to a 5.5 in.b7.5 in. envelope, fall within the overlapping regions, indicating that these envelopes satisf both constraints. Notice that (, ) satisfies the sstem. This point corresponds to an envelope with no length or width, which does not make sense. Adding constraints specifing minimum and maimum lengths and widths would make the sstem a more realistic model. For eample, for an envelope to require a stamp, the length must be between 5 in. and 11.5 in. and the width must be between.5 in. and 6.15 in. The sstem includes these constraints and has this graph. l.5w l 1.w l 5 w.5 l 11.5 w 6.15 Length (in.) l w.5 Length (in.) 5 l c. b. w Width (in.) a. d. 5 Width (in.) w w l 11.5 l Width (in.) w 88 Discovering Algebra Condensed Lessons Ke Curriculum Press
Translating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationLinear Inequalities, Systems, and Linear Programming
8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed
More informationRising Algebra 2 Honors
Rising Algebra Honors Complete the following packet and return it on the first da of school. You will be tested on this material the first week of class. 50% of the test grade will be completion of this
More informationLinear Equations and Arithmetic Sequences
CONDENSED LESSON 3.1 Linear Equations and Arithmetic Sequences In this lesson ou will write eplicit formulas for arithmetic sequences write linear equations in intercept form You learned about recursive
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSolution of the System of Linear Equations: any ordered pair in a system that makes all equations true.
Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes
More informationSection 7.1 Graphing Linear Inequalities in Two Variables
Section 7.1 Graphing Linear Inequalities in Two Variables Eamples of linear inequalities in two variables include + 6, and 1 A solution of a linear inequalit is an ordered pair that satisfies the
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationUnit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system
Unit Stud Guide Sstems of Linear Equations and Inequalities 6 Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and  in for in
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More informationEssential Question How can you graph a system of linear inequalities?
5.7 Sstems of Linear Inequalities Essential Question How can ou graph a sstem of linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More informationy y y 5
Sstems of Linear Inequalities SUGGESTED LEARNING STRATEGIES: Marking the Tet, Quickwrite, Create Representations. Graph each inequalit on the number lines and grids provided. M Notes ACTIVITY.7 Inequalit
More informationInequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,
Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 7577 Es. 420. 2328, 2939 odd, 4043, 4952, 5973 odd
More informationSystems of linear equations (simultaneous equations)
Before starting this topic ou should review how to graph equations of lines. The link below will take ou to the appropriate location on the Academic Skills site. http://www.scu.edu.au/academicskills/numerac/inde.php/1
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationEssential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.
Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationSection 0.2 Set notation and solving inequalities
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
More information3.1 Graphically Solving Systems of Two Equations
3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationAlgebra 2 Unit 1 Practice
Algebra Unit Practice Lesson  Use this information for Items. Aaron has $ to rent a bike in the cit. It costs $ per hour to rent a bike. The additional fee for a helmet is $ for the entire ride.. Write
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationEXAMPLE A Evaluate the expression without a calculator. Then, enter the expression into your calculator to see if you get the same answer.
CONDENSED L E S S O N 4.1 Order of Operations and the Distributive Property In this lesson you will apply the order of operations to evaluate epressions use the distributive property to do mental math
More informationSolving Systems Using Tables and Graphs
 Think About a Plan Solving Sstems Using Tables and Graphs Sports You can choose between two tennis courts at two universit campuses to learn how to pla tennis. One campus charges $ per hour. The other
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationAlgebra 2 Honors: Quadratic Functions. Student Focus
Resources: SpringBoard Algebra Online Resources: Algebra Springboard Tet Algebra Honors: Quadratic Functions Semester 1, Unit : Activit 10 Unit Overview In this unit, students write the equations of quadratic
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More information12.2 Graphing Systems of Linear Inequalities
Name Class Date 1. Graphing Sstems of Linear Inequalities Essential Question: How do ou solve a sstem of linear inequalities? Resource Locker Eplore Determining Solutions of Sstems of Linear Inequalities
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More informationAnswers to Algebra 1 Unit 3 Practice
1. C(t) 5 13t 1 39. $13/da; each etra da the canoe is rented increases the total cost b $13. 3. B. Disagree; the average rate of change between ears 1,1,1 and is 5 388 5 19, while the average rate of change
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationGRAPHING SYSTEMS OF LINEAR INEQUALITIES
444 (8 5) Chapter 8 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationSimultaneous. linear equations. and inequations
Simultaneous linear equations and inequations A Graphical solution of simultaneous linear equations B Solving simultaneous linear equations using substitution C Solving simultaneous linear equations using
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationGraphing a System of Linear Inequalities. To graph a system of linear inequalities, follow these steps:
3.3 Graph Sstems of Linear Inequalities Before You graphed linear inequalities. Now You will graph sstems of linear inequalities. Wh? So ou can model heart rates during eercise, as in E. 39. Ke Vocabular
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationChapter 10 Correlation and Regression. Overview. Section 102 Correlation Key Concept. Definition. Definition. Exploring the Data
Chapter 10 Correlation and Regression 101 Overview 102 Correlation 10 Regression Overview This chapter introduces important methods for making inferences about a correlation (or relationship) between
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationSolving x < a. Section 4.4 Absolute Value Inequalities 391
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
More informationRational Functions, Equations, and Inequalities
Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationSummer Review For Students Entering Algebra 2
Summer Review For Students Entering Algebra Board of Education of Howard Count Frank Aquino Chairman Ellen Flnn Giles Vice Chairman Larr Cohen Allen Der Sandra H. French Patricia S. Gordon Janet Siddiqui
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationAlgebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:
Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving sstems of linear inequalities, including realworld problems through graphing two and three variables
More informationChapter 3: Section 32 Graphing Linear Inequalities
Chapter : Section Graphing Linear Inequalities D. S. Malik Creighton Universit, Omaha, NE D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section Graphing Linear Inequalities / 9 Geometric Approach
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationSYSTEMS OF LINEAR EQUATIONS
SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations
More informationAnswers (Lesson 31) Study Guide and Intervention. Study Guide and Intervention (continued) Solving Systems of Equations by Graphing
Glencoe/McGrawHill A Glencoe Algebra  NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations b Graphing Graph Sstems of Equations A sstem of equations is a set of two or more equations
More informationLesson 8.3 Exercises, pages
Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5  b) = 5 c) =  d) =. Sketch broken lines to represent the vertical and horizontal
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More information10.2 THE QUADRATIC FORMULA
10. The Quadratic Formula (10 11) 535 100. Eploration. Solve k 0 for k 0,, 5, and 10. a) When does the equation have only one solution? b) For what values of k are the solutions real? c) For what values
More informationAre You Ready? Circumference and Area of Circles
SKILL 39 Are You Read? Circumference and Area of Circles Teaching Skill 39 Objective Find the circumference and area of circles. Remind students that perimeter is the distance around a figure and that
More informationChapter 5.1 Systems of linear inequalities in two variables.
Chapter 5. Systems of linear inequalities in two variables. In this section, we will learn how to graph linear inequalities in two variables and then apply this procedure to practical application problems.
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationIntroduction  Algebra I
LIFORNI STNRS TEST lgebra I Introduction  lgebra I The following released test questions are taken from the lgebra I Standards Test. This test is one of the alifornia Standards Tests administered as part
More informationUNCORRECTED PAGE PROOFS
number and algebra ToPIC Simultaneous linear equations and inequalities. Overview Wh learn this? Picture this ou own a factor that produces two different products, and ou are planning to bu some new machines.
More information{ } Sec 3.1 Systems of Linear Equations in Two Variables
Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination
More information3.4 The PointSlope Form of a Line
Section 3.4 The PointSlope Form of a Line 293 3.4 The PointSlope Form of a Line In the last section, we developed the slopeintercept form of a line ( = m + b). The slopeintercept form of a line is
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationSimplifying Algebraic Expressions (pages )
A Simplifing Algebraic Expressions (pages 469 473) Simplifing Algebraic Expressions The expressions 3(x 4) and 3x are equivalent expressions, because no matter what x is, these expressions have the same
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationOrdered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.
Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationS2 Topic 3. Simultaneous Linear Equations. Level: Key Stage 3
Simultaneous Linear Equations S2 Topic 3 Level: Ke Stage 3 Dimension: Number and Algebra Module: Algebraic Relations and Functions Unit: Linear Equations in Two Unknowns Student abilit: Average Content
More informationGraphing Linear Equations in SlopeIntercept Form
4.4. Graphing Linear Equations in SlopeIntercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope
More informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
More informationExponential Functions
CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
More informationMULTIPLE REPRESENTATIONS through 4.1.7
MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects
More informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationMATH chapter 1 Name and Section Number (10 points) Each question below is 10 points unless otherwise noted. Find the slope of the line.
MATH 1  chapter 1 Name and Section Number ( points) Each question below is points unless otherwise noted. Find the slope of the line. 1) Solve the problem. 8) The change in a certain engineer's salar
More informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More information