The graphs intersect. Therefore, there is one solution. The. The solution is (3, 1). many solutions.

Transcription

1 Answers (Lesson 7-) Lesson 7-7- NAME DATE PERID Stud Guide and Intervention Graphing Sstems of Equations Number of Solutions Two or more linear equations involving the same variables form a sstem of equations. A solution of the sstem of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about sstems of linear equations. Graph of a Sstem intersecting lines same line parallel lines Number of Solutions eactl one solution infinitel man solutions no solution consistent and consistent and Terminolog inconsistent independent dependent Eample Use the graph at the right to determine whether the sstem has no solution, one solution, or infinitel man solutions. a. 2 Since the graphs of 2 and intersect, there is one solution. b Since the graphs of 2 and are parallel, there are no solutions c Since the graphs of and coincide, there are infinitel man solutions. Eercises Use the graph at the right to determine whether each sstem has no solution, one solution, or infinitel man solutions one infinitel man none one Glencoe/McGraw-Hill 403 Glencoe Algebra 7- NAME DATE PERID Stud Guide and Intervention (continued) Graphing Sstems of Equations Solve b Graphing ne method of solving a sstem of equations is to graph the equations on the same coordinate plane. Eample Graph each sstem of equations. Then determine whether the sstem has no solution, one solution, or infinitel man solutions. If the sstem has one solution, name it. a. 2 4 The graphs intersect. Therefore, there is one solution. The point (3, ) seems to lie on both lines. Check this estimate b replacing with 3 and with in each equation. 2 3 ( ) ( ) 3 or 4 The solution is (3, ). b The graphs coincide. Therefore there are infinitel man solutions. (3, ) Eercises Graph each sstem of equations. Then determine whether the sstem has no solution, one solution, or infinitel man solutions. If the sstem has one solution, name it.. 2 one; (, 2) 2. 2 one; (2, 3) 3. one; (2, ) (, 2) (2, ) (2, 3) one; (, 4) no solution infinitel man (, 4) Glencoe/McGraw-Hill 404 Glencoe Algebra Glencoe/McGraw-Hill A2 Glencoe Algebra

2 Answers (Lesson 7-) 7- NAME DATE PERID Skills Practice Graphing Sstems of Equations Glencoe/McGraw-Hill 405 Glencoe Algebra Answers Lesson 7- Use the graph at the right to determine whether each sstem has no solution, one solution, or infinitel man solutions infinitel one 4 man no solution one Graph each sstem of equations. Then determine whether the sstem has no solution, one solution, or infinitel man solutions. If the sstem has one solution, name it no 3 one; (, 3) 2 4 one; (, 2) 3 3 solution (, 2) (, 3) infinitel one; 0. 2 man 3 3 ( 3, 0) 3 one; (2, ) (2, ) ( 3, 0) infinitel no solution 2 3 one; (3, 0) 2 2 man (3, 0) NAME DATE PERID 7- Practice (Average) Graphing Sstems of Equations Use the graph at the right to determine whether each sstem has no solution, one solution, or infinitel man solutions no solution infinitel man one 2 3 one Graph each sstem of equations. Then determine whether the sstem has no solution, one solution, or infinitel man solutions. If the sstem has one solution, name it no infinitel one; 3 0 solution man 3 5 (, 2) (, 2) 2 3 BUSINESS For Eercises 8 and 9, use the following information. Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $ Graph the sstem of equations and.5 to represent the situation. 9. How man treats does Nick need to sell per week to break even? Dog Treats (20, 30).5 Sales ($) SALES For Eercises 0 2, use the following information. A used book store also started selling used CDs and videos. In the first week, the store sold 40 used CDs and videos, at $4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $ Write a sstem of equations to represent the situation.. Graph the sstem of equations. c v 40 4c 6v How man CDs and videos did the store sell in the first week? 30 CDs and 0 videos CD and Video Sales c v 40 4c 6v 80 CD Sales ($) (30, 0) Video Sales ($) Cost ($) Glencoe/McGraw-Hill 406 Glencoe Algebra Glencoe/McGraw-Hill A3 Glencoe Algebra

3 Answers (Lesson 7-) Lesson 7-7- NAME DATE PERID Reading to Learn Mathematics Graphing Sstems of Equations Pre-Activit How can ou use graphs to compare the sales of two products? Read the introduction to Lesson 7- at the top of page 369 in our tetbook. What is meant b the term linear function? Sample answer: a function whose graph is a line What does it mean to sa that two lines intersect? Sample answer: The lines cross. Reading the Lesson. Each figure shows the graph of a sstem of two equations. Write the letter of the figures that illustrate each statement. A. B. C. D. a. A sstem of two linear equations can have an infinite number of solutions. D b. A sstem of equations is consistent if there is at least one ordered pair that satisfies both equations. B, C, D c. If two graphs are parallel, there are no ordered pairs that satisf both equations. A d. If a sstem of equations has eactl one solution, it is independent. B, C e. If a sstem of equations has an infinite number of solutions, it is dependent. D Helping You Remember 2. Describe how ou can solve a sstem of equations b graphing. Sample answer: Graph the equations on the same coordinate plane. Locate an points of intersection. Glencoe/McGraw-Hill 407 Glencoe Algebra 7- NAME DATE PERID Enrichment Graphing a Trip The distance formula, d rt, is used to solve man tpes of problems. If ou graph an equation such as d 50t, the graph is a model for a car going at 50 mi/h. The time the car travels is t; the distance in miles the car covers is d. The slope of the line is the speed. Suppose ou drive to a nearb town and return. You average 50 mi/h on the trip out but onl 25 mi/h on the trip home. The round trip takes 5 hours. How far awa is the town? The graph at the right represents our trip. Notice that the return trip is shown with a negative slope because ou are driving in the opposite direction. 50 d 2 slope is 50 slope is 25 t Solve each problem.. Estimate the answer to the problem in the above eample. About how far awa is the town? about 80 miles 2. Graph this trip and solve the problem. An airplane has enough fuel for 3 hours of safe fling. n the trip out the pilot averages 200 mi/h fling against a headwind. n the trip back, the pilot averages 250 mi/h. How long a trip out can the pilot make? 2 about 3 hours and 330 miles 00 d t 3. Graph this trip and solve the 4. Graph this trip and solve the problem. You problem. You drive to a town drive at an average speed of 50 mi/h to a 00 miles awa. n the trip out ou discount shopping plaza, spend 2 hours average 25 mi/h. n the trip back ou shopping, and then return at an average average 50 mi/h. How man hours do speed of 25 mi/h. The entire trip takes ou spend driving? 8 hours. How far awa is the shopping plaza? 6 hours 00 miles d d t 2 t Glencoe/McGraw-Hill 408 Glencoe Algebra Glencoe/McGraw-Hill A4 Glencoe Algebra

4 Answers (Lesson 7-2) 7-2 NAME DATE PERID Stud Guide and Intervention Substitution Glencoe/McGraw-Hill 409 Glencoe Algebra Answers Lesson 7-2 Substitution ne method of solving sstems of equations is substitution. Eample Eample 2 Use substitution to solve the sstem of equations Substitute 2 for in the second equation. 4 4 Second equation Combine like terms. 2 4 Divide each side b Simplif. Use 2 to find the value of. 2 First equation 2( 2) 2 4 Simplif. The solution is ( 2, 4). substitute Solve for one variable, then Solve the first equation for since the coefficient of is. 3 7 First equation Subtract 3 from each side. 7 3 Simplif. Find the value of b substituting 7 3 for in the second equation Second equation 2(7 3) Distributive Propert Combine like terms Subtract 4 from each side Simplif Divide each side b Simplif. Use 2 to find the value of (2) The solution is (, 2). Eercises Use substitution to solve each sstem of equations. If the sstem does not have eactl one solution, state whether it has no solution or infinitel man solutions (, 4) 2 (4, 2) 2 4 no solution c 4d infinitel (5, 3) 2c 8d 2 man (4, 2) 7. 2b 6a 4 infinitel no 3a b 7 man no solution solution 0. 2 (20, 0) ( 3, ) 0.4. (, 0.7) 7-2 NAME DATE PERID Stud Guide and Intervention (continued) Substitution Real-World Problems Substitution can also be used to solve real-world problems involving sstems of equations. It ma be helpful to use tables, charts, diagrams, or graphs to help ou organize data. Eample CHEMISTRY How much of a 0% saline solution should be mied with a 20% saline solution to obtain 000 milliliters of a 2% saline solution? Let s the number of milliliters of 0% saline solution. Let t the number of milliliters of 20% saline solution. Use a table to organize the information. 0% saline 20% saline 2% saline Total milliliters s t 000 Milliliters of saline 0.0s 0.20t 0.2(000) Write a sstem of equations. s t s 0.20t 0.2(000) Use substitution to solve this sstem. s t 000 First equation s 000 t Solve for s. 0.0s 0.20t 0.2(000) Second equation 0.0(000 t) 0.20t 0.2(000) s 000 t t 0.20t 0.2(000) Distributive Propert t 0.2(000) Combine like terms. 0.0t 20 Simplif. 0.0t 20 Divide each side b t 200 Simplif. s t 000 First equation s t 200 s 800 Solve for s. 800 milliliters of 0% solution and 200 milliliters of 20% solution should be used. Eercises. SPRTS At the end of the football season, 3 Super Bowl games had been plaed with the current two football leagues, the American Football Conference (AFC) and the National Football Conference (NFC). The NFC won five more games than the AFC. How man games did each conference win? Source: New York Times Almanac AFC 3; NFC 8 2. CHEMISTRY A lab needs to make 00 gallons of an 8% acid solution b miing a 2% acid solution with a 20% solution. How man gallons of each solution are needed? 25 gal of 2% solution and 75 gal of 20% solution 3. GEMETRY The perimeter of a triangle is 24 inches. The longest side is 4 inches longer than the shortest side, and the shortest side is three-fourths the length of the middle side. Find the length of each side of the triangle. 6 in., 8 in., 0 in. Glencoe/McGraw-Hill 40 Glencoe Algebra Glencoe/McGraw-Hill A5 Glencoe Algebra

5 Answers (Lesson 7-2) Lesson NAME DATE PERID Skills Practice Substitution Use substitution to solve each sstem of equations. If the sstem does not have eactl one solution, state whether it has no solution or infinitel man solutions (, 4) 3 4 ( 2, 4) (3, 9) (8, 2) (2, ) 8 2 ( 6, ) ( 2, 9) (, 5) (6, 3) (3, 7) (7, 3) 3 5 no solution (3, 5) (2, ) infinitel man 2 5 (4, 3) (9, 4) 3 23 ( 5, 8) , ( 2, 3) Glencoe/McGraw-Hill 4 Glencoe Algebra NAME DATE PERID 7-2 Practice (Average) Substitution Use substitution to solve each sstem of equations. If the sstem does not have eactl one solution, state whether it has no solution or infinitel man solutions (, 6) (9, 3) 4 (, 3) (4, 6) 2 2 no solution 2 (7, 9) ( 3, 8) infinitel ( 4, 8) man ( 6, 4) (4, 5) 2 5 (5, 5) (6, ) 2 4 (5, 2) 2 6 (2, 3) (2, ),, EMPLYMENT For Eercises 9 2, use the following information. Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4% commission on her total sales, or $400 per month plus a 5% commission on total sales. 9. Write a sstem of equations to represent the situation and What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pa scale? $0, Which is the better offer? the first offer if she epects to sell less than $0,000 in shoes, and the second offer if she epects to sell more than $0,000 in shoes MVIE TICKETS For Eercises 22 and 23, use the following information. Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased 8 tickets for $ Write a sstem of equations to represent the situation. 8 and How man adult tickets and student tickets were purchased? 5 adult and 3 student Glencoe/McGraw-Hill 42 Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra

6 Answers (Lesson 7-2) 7-2 NAME DATE PERID Reading to Learn Mathematics Substitution Glencoe/McGraw-Hill 43 Glencoe Algebra Answers Lesson 7-2 Pre-Activit How can a sstem of equations be used to predict media use? Read the introduction to Lesson 7-2 at the top of page 376 in our tetbook. What is the sstem of equations? , Based on the graph, are there 0,, or infinitel man solutions of the sstem? solution Reading the Lesson. Describe how ou would use substitution to solve each sstem of equations. a Substitute 2 for in the second equation. Simplif and solve for. Then use the value of to find the value of. b Substitute 2 for in the first equation. Simplif and solve for. Then use the value of to find the value of. c Solve the first equation for. Substitute the epression ou find for in the second equation. Simplif and solve for. Then use the value of to find the value of. d Solve the second equation for. Substitute the epression ou find for in the first equation. Simplif and solve for. Then use the value of to find the value of. 2. Jess solved a sstem of equations and her result was 8 8. All of her work was correct. Describe the graph of the sstem. Eplain. It is a line. There are infinitel man solutions, since 8 8 is alwas true. 3. Miguel solved a sstem of equations and his result was 5 2. All of his work was correct. Describe the graph of the sstem. Eplain. The graph has two parallel lines, since 5 2 is alwas false. Helping You Remember 4. What is usuall the first step in solving a sstem of equations b substitution? Sample answer: Solve one of the equations for one variable in terms of the other. 7-2 NAME DATE PERID Enrichment Equations of Lines and Planes in Intercept Form ne form that a linear equation ma take is intercept form. The constants a and b are the - and -intercepts of the graph. a b z In three-dimensional space, the equation of a plane takes a similar form. a z b c Here, the constants a, b, and c are the points where the plane meets the,, and z-aes. 8 7 z 6 Solve each problem. z. Graph the equation 3 2. z 2. For the plane in Eercise, write an equation for the line where the plane intersects the -plane. Use intercept forms. 3. Write an equation for the line where the plane intersects the z-plane. z 4. Write an equation for the line where the plane intersects the z-plane. z z 5. Graph the equation 4 2. z 6. Write an equation for the -plane. z 0 7. Write an equation for the z-plane Write an equation for a plane parallel to the -plane with a z-intercept of 2. z 2 9. Write an equation for a plane parallel to the z-plane with an -intercept of Glencoe/McGraw-Hill 44 Glencoe Algebra Glencoe/McGraw-Hill A7 Glencoe Algebra

7 Answers (Lesson 7-3) Lesson NAME DATE PERID Stud Guide and Intervention Elimination Using Addition and Subtraction Elimination Using Addition In sstems of equations in which the coefficients of the or terms are additive inverses, solve the sstem b adding the equations. Because one of the variables is eliminated, this method is called elimination. Use addition to solve the sstem of equations Write the equations in column form and add to eliminate. 3 7 ( ) Solve for. 4 4 Eample Eample Substitute 4 for in either equation and solve for The solution is (4, ). The sum of two numbers is 70 and their difference is 24. Find the numbers. Let represent one number and represent the other number. 70 ( ) Substitute 47 for in either equation The numbers are 47 and 23. Eercises Use elimination to solve each sstem of equations m 3n a b 9 2 (, 3) m 3n (, 4) 3a 2b 0 ( 2, 3) (, ) 2c d 6 (2, 2) 2 6 2, c d (2, 3) 4 4 2, 3 2, (6, 0) (, 0.7) , Rema is older than Ken. The difference of their ages is 2 and the sum of their ages is 50. Find the age of each. Rema is 3 and Ken is The sum of the digits of a two-digit number is 2. The difference of the digits is 2. Find the number if the units digit is larger than the tens digit. 57 Glencoe/McGraw-Hill 45 Glencoe Algebra 7-3 NAME DATE PERID Stud Guide and Intervention (continued) Elimination Using Addition and Subtraction Elimination Using Subtraction In sstems of equations where the coefficients of the or terms are the same, solve the sstem b subtracting the equations. Eample Use subtraction to solve the sstem of equations. 2 3 Write the equations in column form and subtract. ( ) Subtract the two equations. is eliminated Divide each side b 3. 3 Simplif. 2() 3 Substitute for in either equation. 2 3 Simplif Subtract 2 from each side. 3 9 Simplif Divide each side b Simplif. The solution is (, 3). Eercises Use elimination to solve each sstem of equations m 4n a b m 2n 2 a b 3 (, 2) ( 2, 2) (, 4) c 3d c 3d 6 3 (, 5) (5, 2) (4, ) 7. 2a 3b s t 6 2a 2b s 2t 3 2 2, 4, 2 (, ) ( 6, 6) (.5, 2.5) (, 0.5) 3. The sum of two numbers is 70. ne number is ten more than twice the other number. Find the numbers. 50; GEMETRY Two angles are supplementar. The measure of one angle is 0 more than three times the other. Find the measure of each angle ; 37.5 Glencoe/McGraw-Hill 46 Glencoe Algebra Glencoe/McGraw-Hill A8 Glencoe Algebra

8 Answers (Lesson 7-3) 7-3 NAME DATE PERID Skills Practice Elimination Using Addition and Subtraction Glencoe/McGraw-Hill 47 Glencoe Algebra Answers Lesson 7-3 Use elimination to solve each sstem of equations (2, ) (5, 6) (, 0) 3 8 (6, 4) ( 3, 7) 4 8 ( 8, 0) 7. 3a 4b c d 4a 4b 2 (2, ) 3c d 5 (, 2) ( 3, 5) 2 4 (0, 4). 3m n m n 24 (0, 4) 2 (, ) (7, 5) ( 5, 2) 5. 7m 4n m 2n 8 (2, 3) ( 4, 4) 7. The sum of two numbers is 28 and their difference is 4. What are the numbers? 2, 6 8. Find the two numbers whose sum is 29 and whose difference is 5. 7, The sum of two numbers is 24 and their difference is 2. What are the numbers? 3, 20. Find the two numbers whose sum is 54 and whose difference is 4. 25, Two times a number added to another number is 25. Three times the first number minus the other number is 20. Find the numbers. 9, 7 NAME DATE PERID 7-3 Practice (Average) Elimination Using Addition and Subtraction Use elimination to solve each sstem of equations.. 2. p q p q ( 4, 5) (3, 5) (5, 3) (6, 3) ( 5, 7) (2, 4) c 2d c 2d 4 (3, 4) ( 7, ) ( 2, 5) (9, 2) ( 2, 8) (.6, 2.5) m 8n m 8n 3 (2.5,.25) (3.4, 2.2) 2, a b a 3b 0 2, (0, ) (2, 2) The sum of two numbers is 4 and their difference is 5. What are the numbers? 8, Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the numbers. 8, 4 2. ne number added to three times another number is 24. Five times the first number added to three times the other number is 36. Find the numbers. 3, LANGUAGES English is spoken as the first or primar language in 78 more countries than Farsi is spoken as the first language. Together, English and Farsi are spoken as a first language in 30 countries. In how man countries is English spoken as the first language? In how man countries is Farsi spoken as the first language? English: 04 countries, Farsi: 26 countries 23. DISCUNTS At a sale on winter clothing, Cod bought two pairs of gloves and four hats for $ Tori bought two pairs of gloves and two hats for $ What were the prices for the gloves and hats? gloves: $8.50, hats: $6.50. Glencoe/McGraw-Hill 48 Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

9 Answers (Lesson 7-3) Lesson NAME DATE PERID Reading to Learn Mathematics Elimination Using Addition and Subtraction Pre-Activit How can ou use a sstem of equations to solve problems about weather? Read the introduction to Lesson 7-3 at the top of page 382 in our tetbook. What fact eplains wh the variable d gets eliminated from the sstem of equations? d d 0 Reading the Lesson. Write addition or subtraction to tell which operation it would be easiest to use to eliminate a variable of the sstem. Eplain our choice. Sstem of Equations peration Eplanation a addition The coefficients of the terms are additive inverses. b subtraction The coefficients of the terms are the same. c. 4 9 subtraction The coefficients of the terms are the same. d addition The coefficients of the terms are additive inverses. Helping You Remember 2. Tell how ou can decide whether to use addition or subtraction to eliminate a variable in a sstem of equations. Sample answer: Look at the coefficients of each variable. If the coefficients of one of the variables are additive inverses, ou can use addition to eliminate the variable. If the coefficients of one variable are the same, ou can use subtraction to eliminate the variable. Glencoe/McGraw-Hill 49 Glencoe Algebra 7-3 NAME DATE PERID Enrichment Rózsa Péter Rózsa Péter ( ) was a Hungarian mathematician dedicated to teaching others about mathematics. As professor of mathematics at a teachers college in Budapest, she wrote several mathematics tetbooks and championed reforms in the teaching of mathematics. In 945 she wrote Plaing with Infinit: Mathematical Eplorations and Ecursions, a popular work in which she attempted to conve the spirit of mathematics to the general public. B far Péter s greatest contribution to mathematics was her pioneering research in the field of recursive function theor. When ou evaluate a function recursivel, ou begin with one initial value of. Working from this single number, ou can use the function to generate an entire sequence of numbers. For instance, here is how ou use an initial value of to evaluate the function f() 3 recursivel. f() 3() 3 f(3) 3(3) 9 f(9) 3(9) 27 f(27) 3(27) 8 f(8) 3(8) 243 The first five numbers of the sequence generated b this function are 3, 9, 27, 8, and 243. Write the first five numbers of the sequence generated b each function, using the given number as the initial value of.. f() 3; 2 2. g() 5; 6, 8, 54, 62, 486 4, 9, 4, 9, f() 2 ; 3 4. f() 2 ; 2 5, 9, 7, 33, 65 4; 6; 256; 65,536; 4,294,967, h() ; 3 6. k() ; 0 3, 3, 3, 3, 3 0, 0, 0, 0, 0 Glencoe/McGraw-Hill 420 Glencoe Algebra Glencoe/McGraw-Hill A0 Glencoe Algebra

10 Answers (Lesson 7-4) 7-4 NAME DATE PERID Stud Guide and Intervention Elimination Using Multiplication Glencoe/McGraw-Hill 42 Glencoe Algebra Answers Lesson 7-4 Elimination Using Multiplication Some sstems of equations cannot be solved simpl b adding or subtracting the equations. In such cases, one or both equations must first be multiplied b a number before the sstem can be solved b elimination. Eample Eample 2 Use elimination to solve the sstem of equations If ou multipl the second equation b 2, ou can eliminate the terms. 0 3 ( ) Substitute for in either equation The solution is, Use elimination to solve the sstem of equations If ou multipl the first equation b 2 and the second equation b 3, ou can eliminate the terms ( ) Substitute 4 for in either equation. 3 2( 4) The solution is ( 5, 4). Eercises Use elimination to solve each sstem of equations m 3n a b ( 3, 4) m 2n 5 (, 2) a 2b 3 (, ) c 3d (, 2) 2c d 0 (4, 2) 3 0 ( 4, 2) 7. 4s t a 3b s 2t 8 (2, ) 2a 2b 3, , (0.9,.7) , 2 4, GARDENING The length of Sall s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 72 meters. What are the dimensions of Sall s garden? 28 m b 8 m 4. Anita is 4 2 ears older than Basilio. Three times Anita s age added to si times Basilio s age is 36. How old are Anita and Basilio? Anita: 7 r; Basilio: 2 2 r 7-4 NAME DATE PERID Stud Guide and Intervention (continued) Elimination Using Multiplication Determine the Best Method The methods to use for solving sstems of linear equations are summarized in the table below. Method The Best Time to Use Graphing to estimate the solution, since graphing usuall does not give an eact solution Substitution if one of the variables in either equation has a coefficient of or Elimination Using Addition if one of the variables has opposite coefficients in the two equations Elimination Using Subtraction if one of the variables has the same coefficient in the two equations Elimination Using Multiplication if none of the coefficients are or and neither of the variables can be eliminated b simpl adding or subtracting the equations Eample Determine the best method to solve the sstem of equations. Then solve the sstem Since the coefficients of will be additive inverses of each other if ou multipl the second equation b 3, use elimination ( ) Multipl the second equation b Add the two equations. is eliminated Divide each side b Simplif. The solution is (4, 2). 6 2( 2) 20 Substitute 2 for in either equation Simplif Add 4 to each side Simplif Divide each side b Simplif. Eercises Determine the best method to solve each sstem of equations. Then solve the sstem m 6n 8 3. a b 6 m 2n 8 a 2b 7 elimination ( ); (, 2) substitution; (4, 2) substitution; (5, ) c d c d 2 4 substitution; (3, 3) elimination ( ); (6, 4) substitution; (, 4) s 3t s 2t 24 5 substitution; (, 3) substitution; 5, 2 elimination ( ); (2, 6) 0. 4a 4b a 4b elimination ( ); 2, 2 elimination ( ); 2, 5 substitution; ( 3, 3) Glencoe/McGraw-Hill 422 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

11 Answers (Lesson 7-4) Lesson NAME DATE PERID Skills Practice Elimination Using Multiplication Use elimination to solve each sstem of equations (2, ) 3 ( 7, 6) (4, ) (, ) (, 5) ( 2, 4) ( 2, 0) (4, 2) (3, 5) ( 6, 4) (7, 4) ( 3, 6) 3. Two times a number plus three times another number equals 3. The sum of the two numbers is 7. What are the numbers? 8, 4. Four times a number minus twice another number is 6. The sum of the two numbers is. Find the numbers. 3, 2 Determine the best method to solve each sstem of equations. Then solve the sstem elimination ( ); elimination ( ); ( 4, 6) (4, 2) 7. 2 substitution; elimination ( ); (5, 0) 3 3 no solution elimination ( ); substitution; (3, 2) 3 infinitel man solutions Glencoe/McGraw-Hill 423 Glencoe Algebra NAME DATE PERID 7-4 Practice (Average) Elimination Using Multiplication Use elimination to solve each sstem of equations ( 3, 5) (2, 0) ( 4, 6) (3, 7) (, 3) (7, 2) (5, 3) (4, 8) 2 0 ( 2, 4) 0. Eight times a number plus five times another number is 3. The sum of the two numbers is. What are the numbers? 6, 7. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers. 5, 2 Determine the best method to solve each sstem of equations. Then solve the sstem elimination ( ); elimination ( ); elimination ( ); ( 5, 4) ( 2, 8) ( 2, ) substitution; infinitel elimination ( ); man solutions no solution substitution; (6, 3) 8. FINANCE Gunther invested $0,000 in two mutual funds. ne of the funds rose 6% in one ear, and the other rose 9% in one ear. If Gunther s investment rose a total of $684 in one ear, how much did he invest in each mutual fund? $7200 in the 6% fund and $2800 in the 9% fund 9. CANEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at which Laura and Brent paddled the canoe in still water..75 mi/h 20. NUMBER THERY The sum of the digits of a two-digit number is. If the digits are reversed, the new number is 45 more than the original number. Find the number. 38 Glencoe/McGraw-Hill 424 Glencoe Algebra Glencoe/McGraw-Hill A2 Glencoe Algebra

12 Answers (Lesson 7-4) 7-4 NAME DATE PERID Reading to Learn Mathematics Elimination Using Multiplication Glencoe/McGraw-Hill 425 Glencoe Algebra Answers Lesson 7-4 Pre-Activit How can a manager use a sstem of equations to plan emploee time? Read the introduction to Lesson 7-4 at the top of page 387 in our tetbook. Can the sstem of equations be solved b elimination with addition or subtraction? Eplain. No; neither variable has coefficients that are additive inverses or equal. Reading the Lesson. Could elimination b multiplication be used to solve the sstem shown below? Eplain Yes; ou can multipl the first equation b 2 to make the coefficients of the terms additive inverses. 2. Tell whether it would be easiest to use substitution, elimination b addition, elimination b subtraction, or elimination b multiplication to solve the sstem. Eplain our choice. Sstem of Equations Solution Method Eplanation a elimination b The coefficients of the terms are addition additive inverses. b. 2 0 substitution It is eas to solve the first equation for. c elimination b Sample answer: You can multipl the multiplication first equation b 2 to eliminate b addition. d elimination b The coefficients of the terms are subtraction the same. Helping You Remember 3. If ou are going to solve a sstem b elimination, how do ou decide whether ou will need to multipl one or both equations b a number? Sample answer: If both adding the equations and subtracting the equations result in equations that still have two variables, ou will need to use multiplication. 7-4 NAME DATE PERID Enrichment George Washington Carver and Perc Julian In 990, George Washington Carver and Perc Julian became the first African Americans elected to the National Inventors Hall of Fame. Carver ( ) was an agricultural scientist known worldwide for developing hundreds of uses for the peanut and the sweet potato. His work revitalized the econom of the southern United States because it was no longer dependent solel upon cotton. Julian ( ) was a research chemist who became famous for inventing a method of making a snthetic cortisone from sobeans. His discover has had man medical applications, particularl in the treatment of arthritis. There are dozens of other African American inventors whose accomplishments are not as well known. Their inventions range from common household items like the ironing board to comple devices that have revolutionized manufacturing. The eercises that follow will help ou identif just a few of these inventors and their inventions. Match the inventors with their inventions b matching each sstem with its solution. (Not all the solutions will be used.). Sara Boone 2 E A. (, 4) automatic traffic signal 0 2. Sarah Goode 2 D B. (4, 2) eggbeater Frederick M. 2 6 G C. ( 2, 3) fire etinguisher Jones 3 4. J. L. Love F D. ( 5, 7) folding cabinet bed T. J. Marshall 3 9 C E. (6, 4) ironing board Jan Matzeliger 4 2 J F. ( 2, 4) pencil sharpener Garrett A A G. ( 3, 0) portable X-ra machine Morgan Norbert Rillieu 3 2 I H. (2, 3) plaer piano 3 5 I. no solution evaporating pan for refining sugar J. infinitel lasting (shaping) man machine for solutions manufacturing shoes Glencoe/McGraw-Hill 426 Glencoe Algebra Glencoe/McGraw-Hill A3 Glencoe Algebra

13 Answers (Lesson 7-5) Lesson NAME DATE PERID Stud Guide and Intervention Graphing Sstems of Inequalities Sstems of Inequalities The solution of a sstem of inequalities is the set of all ordered pairs that satisf both inequalities. If ou graph the inequalities in the same coordinate plane, the solution is the region where the graphs overlap. Eample b graphing. 2 2 Solve the sstem of inequalities The solution includes the ordered pairs in the intersection of the graphs. This region is shaded at the right. The graphs of 2 and 2 are boundaries of this region. The graph of 2 is dashed and is not included in the graph of Eample 2 b graphing. 4 Solve the sstem of inequalities The graphs of 4 and are parallel. Because the two regions have no points in common, the sstem of inequalities has no solution. 4 Eercises Solve each sstem of inequalities b graphing Glencoe/McGraw-Hill 427 Glencoe Algebra 7-5 NAME DATE PERID Stud Guide and Intervention (continued) Graphing Sstems of Inequalities Real-World Problems In real-world problems, sometimes onl whole numbers make sense for the solution, and often onl positive values of and make sense. Eample BUSINESS AAA Gem Compan produces necklaces and bracelets. In a 40-hour week, the compan has 400 gems to use. A necklace requires 40 gems and a bracelet requires 0 gems. It takes 2 hours to produce a necklace and a bracelet requires one hour. How man of each tpe can be produced in a week? Let n the number of necklaces that will be produced and b the number of bracelets that will be produced. Neither n or b can be a negative number, so the following sstem of inequalities represents the conditions of the problems Necklaces n 0 b 0 b 2n 40 0b 40n 400 The solution is the set ordered pairs in the intersection of the graphs. This region is shaded at the right. nl whole-number solutions, such as (5, 20) make sense in this problem. For each eercise, graph the solution set. List three possible solutions to the problem.. HEALTH Mr. Flowers is on a restricted 2. RECREATIN Maria had $50 in gift diet that allows him to have between certificates to use at a record store. She 600 and 2000 Calories per da. His bought fewer than 20 recordings. Each dail fat intake is restricted to between tape cost $5.95 and each CD cost $ and 55 grams. What dail Calorie How man of each tpe of recording might and fat intakes are acceptable? she have bought? t c t 8.95c 50 Compact Discs Fat Grams Bracelets 0 0 0b 40n 400 b 2n 40 Eercises Calories Tapes Sample answers: 600 Calories, Sample answers: 0 tapes, 9 CDs; 45 fat grams; 800 Calories, 50 fat 0 tapes, 6 CDs; 4 tapes, 5 CDs grams; 2000 Calories, 55 fat grams Glencoe/McGraw-Hill 428 Glencoe Algebra Glencoe/McGraw-Hill A4 Glencoe Algebra

14 Answers (Lesson 7-5) 7-5 NAME DATE PERID Skills Practice Graphing Sstems of Inequalities Glencoe/McGraw-Hill 429 Glencoe Algebra Answers Lesson 7-5 Solve each sstem of inequalities b graphing Write a sstem of inequalities for each graph , 3,, NAME DATE PERID 7-5 Practice (Average) Graphing Sstems of Inequalities Solve each sstem of inequalities b graphing FITNESS For Eercises 7 and 8, use the following information. Diego started an eercise program in which each week he works out at the gm between 4.5 and 6 hours and walks between 9 and 2 miles. 7. Make a graph to show the number of hours Diego works out at the gm and the number of miles he walks per week. 8. List three possible combinations of working out and walking that meet Diego s goals. Sample answers: gm 5 h, walk 9 mi; gm 6 h, walk 0 mi, gm 5.5 h, walk mi Diego s Routine Gm (hours) SUVENIRS For Eercises 9 and 0, use the following information. Emil wants to bu turquoise stones on her trip to New Meico to give to at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emil has no more than $30 to spend. 9. Make a graph showing the numbers of each price of stone Emil can purchase. 0. List three possible solutions. Sample answer: one $4 stone and four $6 stones; three $4 stones and three $6 stones; five $4 stones and one $6 stone Turquoise Stones $6 Stones Waking (miles) $4 Stones Glencoe/McGraw-Hill 430 Glencoe Algebra Glencoe/McGraw-Hill A5 Glencoe Algebra

15 Answers (Lesson 7-5) Lesson NAME DATE PERID Reading to Learn Mathematics Graphing Sstems of Inequalities Pre-Activit How can ou use a sstem of inequalities to plan a sensible diet? Read the introduction to Lesson 7-5 at the top of page 394 in our tetbook. The green section on the graph represents a range of 2000 to 2400 Calories a da and 60 to 75 grams of fat per da. Reading the Lesson Write the inequalit smbols that ou need to get a sstem whose graph looks like the one shown. Use,,, or Helping You Remember 5. Describe how ou would eplain the process of using a graph to solve a sstem of inequalities to a friend who missed Lesson 7-5. Graph each inequalit on the same coordinate plane. The solutions are the ordered pairs for the points in both graphs. Glencoe/McGraw-Hill 43 Glencoe Algebra 7-5 NAME DATE PERID Enrichment Describing Regions The shaded region inside the triangle can be described with a sstem of three inequalities Write sstems of inequalities to describe each region. You ma first need to divide a region into triangles or quadrilaterals top: 2 7, 2 7, 4 4 middle: 4, 0, 3 4, bottom left: 4 2, 2 2, 0, 0 bottom right: 4 2, 2 2, 0, 0 Glencoe/McGraw-Hill 432 Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra