Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The solution is (3, 2). Study Guide and Review - Chapter 6 ANSWER: one; (3, 2) Graph each system and determine the number of solutions that it has. If it has one solution, name it. 9. x − y = 1 x +y = 5 SOLUTION: To graph the system, write both equations in slopeintercept form. Equation 1: 10. y = 2x − 4 4x + y = 2 Equation 2: Graph and locate the solution. y=x−1 y = −x + 5 SOLUTION: To graph the system, write both equations in slopeintercept form. Equation 2: Graph and find the solution. y = 2x − 4 y = −4x + 2 The graphs appear to intersect at the point (3, 2). You can check this by substituting 3 for x and 2 for y. The graphs appear to intersect at the point (1, −2). You can check this by substituting 1 for x and −2 for y. The solution is (3, 2). ANSWER: one; (3, 2) eSolutions Manual - Powered by Cognero The solution is (1, −2). ANSWER: one; (1, −2) Page 1 Study Guide and Review - Chapter 6 The solution is (1, −2). ANSWER: one; (1, −2) The graphs appear to intersect at the point (0, 2). You can check this by substituting 0 for x and 2 for y. The solution is (0, 2). ANSWER: one; (0, 2) 11. 2x − 3y = −6 y = −3x + 2 SOLUTION: To graph the system, write both equations in slopeintercept form. Equation 1: 12. −3x + y = −3 y =x−3 SOLUTION: To graph the system, write both equations in slopeintercept form. Graph and find the solution. y= Equation 1: x+2 y = −3x + 2 Graph and find the solution. y = 3x − 3 y=x−3 The graphs appear to intersect at the point (0, 2). You can check this by substituting 0 for x and 2 for y. eSolutions Manual - Powered by Cognero Page 2 The graphs appear to intersect at the point (0, −3). Graph and find the solution. y =Guide 3x − 3and Review - Chapter 6 Study y=x−3 SOLUTION: To graph the system, write both equations in slopeintercept form. Equation 1: Equation 2: The graphs appear to intersect at the point (0, −3). You can check this by substituting 0 for x and −3 for y. Graph each equation. y= x+3 y= x+ The solution is (0, −3). ANSWER: one; (0, −3) 13. x + 2y = 6 3x + 6y = 8 SOLUTION: To graph the system, write both equations in slopeintercept form. The lines have the same slope but different yintercepts, so the lines are parallel. Since they do not intersect, there is no solution of this system. The system is inconsistent. ANSWER: no solution Equation 1: eSolutions Manual - Powered by Cognero Equation 2: Page 3 14. 3x + y = 5 6x = 10 − 2y intercepts, so the lines are parallel. Since they do not intersect, there is no solution of this system. The system is inconsistent. Study Guide and Review - Chapter 6 ANSWER: no solution The two lines are identical, so there are an infinite number of solutions to the system. The system is dependent. ANSWER: infinitely many solutions 14. 3x + y = 5 6x = 10 − 2y SOLUTION: To graph the system, write both equations in slopeintercept form. 15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two variables, write a system of equations, and solve by graphing. SOLUTION: Sample answer: Let x be one number and y be the other number. Equation 2: x + y = 14 x −y = 4 Equation 1: To graph the system, write both equations in slopeintercept form. Graph the equations. y = −3x + 5 y = −3x + 5 Equation 1: Equation 2: Graph the equations and find the solution. The two lines are identical, so there are an infinite number of solutions to the system. The system is dependent. eSolutions Manual - Powered by Cognero ANSWER: infinitely many solutions y = −x + 14 y=x−4 Page 4 Graph the equations and find the solution. y =Guide −x + 14 Study and Review - Chapter 6 y=x−4 Use substitution to solve each system of equations. 16. x + y = 3 x = 2y SOLUTION: x +y = 3 x = 2y Substitute 2y for x in the first equation. The graphs appear to intersect at the point (9, 5). So, the numbers 9 and 5 have a sum of 14 and a difference of 4. ANSWER: Sample answer: Let x be one number and y the other number; x + y = 14; x − y = 4; 9 and 5 Use the solution for y and either equation to find x. x = 2y x = 2(1) x =2 The solution is (2, 1). ANSWER: (2, 1) 17. x + 3y = −28 y = −5x Use substitution to solve each system of equations. 16. x + y = 3 x = 2y SOLUTION: x + 3y = −28 y = −5x Substitute −5x for y in the first equation. SOLUTION: x +y = 3 x = 2y Substitute 2y for x in the first equation. Use the solution for x and either equation to find y. Use the solution for y and either equation to find x. x = 2y x = 2(1) eSolutions x = 2Manual - Powered by Cognero The solution is (2, 1). The solution is (2, −10). ANSWER: (2, −10) Page 5 The solution is (2, 1). ANSWER: Study and Review - Chapter 6 (2,Guide 1) 17. x + 3y = −28 y = −5x SOLUTION: x + 3y = −28 y = −5x Substitute −5x for y in the first equation. The solution is (2, −10). ANSWER: (2, −10) 18. 3x + 2y = 16 x = 3y − 2 SOLUTION: 3x + 2y = 16 x = 3y − 2 Substitute 3y − 2 for x in the first equation. Use the solution for x and either equation to find y. Use the solution for y and either equation to find x. The solution is (2, −10). ANSWER: (2, −10) 18. 3x + 2y = 16 x = 3y − 2 SOLUTION: 3x + 2y = 16 x = 3y − 2 Substitute 3y − 2 for x in the first equation. The solution is (4, 2). ANSWER: (4, 2) 19. x − y = 8 y = −3x SOLUTION: x −y = 8 y = −3x Substitute −3x for y in the first equation. Use the solution for x and either equation to find y. The solution is (4, 2). The solution is (2, −6). ANSWER: ANSWER: Use the solution for y and either equation to find x. eSolutions Manual - Powered by Cognero Page 6 The solution is (4, 2). ANSWER: Study Guide and Review - Chapter 6 (4, 2) 19. x − y = 8 y = −3x The solution is (2, −6). ANSWER: (2, −6) 20. y = 5x − 3 x + 2y = 27 SOLUTION: x −y = 8 y = −3x Substitute −3x for y in the first equation. SOLUTION: y = 5x − 3 x + 2y = 27 Substitute 5x − 3 for y in the second equation. Use the solution for x and either equation to find y. Use the solution for x and either equation to find y. The solution is (2, −6). ANSWER: (2, −6) 20. y = 5x − 3 x + 2y = 27 SOLUTION: y = 5x − 3 x + 2y = 27 Substitute 5x − 3 for y in the second equation. The solution is (3, 12). ANSWER: (3, 12) 21. x + 3y = 9 x +y = 1 SOLUTION: x + 3y = 9 x +y = 1 First, solve the second equation for y to get y = −x +1. Then substitute −x + 1 for y in the first equation. Use the solution for x and either equation to find y. The solution is (3, 12). eSolutions Manual - Powered by Cognero ANSWER: (3, 12) Use the solution for x and either equation to find y. The solution is (−3, 4). ANSWER: (−3, 4) Page 7 The solution is (3, 12). ANSWER: Study Guide and Review - Chapter 6 (3, 12) 21. x + 3y = 9 x +y = 1 SOLUTION: x + 3y = 9 x +y = 1 First, solve the second equation for y to get y = −x +1. Then substitute −x + 1 for y in the first equation. The solution is (−3, 4). ANSWER: (−3, 4) 22. GEOMETRY The perimeter of a rectangle is 48 inches. The length is 6 inches greater than the width. Define the variables, and write equations to represent this situation. Solve the system by using substitution. SOLUTION: Sample answer: Let w be the width and be the length. 2 + 2w = 48 =w+6 Substitute w + 6 for in the first equation. Use the solution for x and either equation to find y. Use the solution for w and either equation to find . The solution is (−3, 4). ANSWER: (−3, 4) 22. GEOMETRY The perimeter of a rectangle is 48 inches. The length is 6 inches greater than the width. Define the variables, and write equations to represent this situation. Solve the system by using substitution. SOLUTION: Sample answer: Let w be the width and be the length. 2 + 2w = 48 =w+6 Substitute w + 6 for in the first equation. =w+6 =9+6 = 15 The solution is (9, 15). ANSWER: Sample answer: Let w be the width and l be the length; 2l + 2w = 48, l = w + 6; 9 is the width and 15 is the length. Use elimination to solve each system of equations. 23. x + y = 13 x −y = 5 SOLUTION: Because y and −y have opposite coefficients, add the equations. Use the solution for w and either equation to find . =w+6 = + 6 - Powered by Cognero eSolutions9Manual = 15 The solution is (9, 15). Now, substitute 9 for x in either equation to find y. Page 8 ANSWER: Sample answer: Let w be the width and l be the Study Guide length; 2l and + 2wReview = 48, l =- Chapter w + 6; 9 is6the width and 15 is the length. Use elimination to solve each system of equations. 23. x + y = 13 x −y = 5 SOLUTION: Because y and −y have opposite coefficients, add the equations. The solution is (9, 4). ANSWER: (9, 4) 24. −3x + 4y = 21 3x + 3y = 14 SOLUTION: Because −3x and 3x have opposite coefficients, add the equations. Now, substitute 9 for x in either equation to find y. Now, substitute 5 for y in either equation to find x. The solution is (9, 4). ANSWER: (9, 4) The solution is 24. −3x + 4y = 21 3x + 3y = 14 . ANSWER: SOLUTION: Because −3x and 3x have opposite coefficients, add the equations. 25. x + 4y = −4 x + 10y = −16 SOLUTION: Because x and x have the same coefficients, multiply equation 1 by –1 so the x's are additive inverses. Then add the equations. Now, substitute 5 for y in either equation to find x. eSolutions Manual - Powered by Cognero The solution is ANSWER: . Now, substitute −2 for y in either equation to find x. Page 9 ANSWER: Study Guide and Review - Chapter 6 25. x + 4y = −4 x + 10y = −16 SOLUTION: Because x and x have the same coefficients, multiply equation 1 by –1 so the x's are additive inverses. Then add the equations. The solution is (4, −2). ANSWER: (4, −2) 26. 2x + y = −5 x −y = 2 SOLUTION: Because y and −y have opposite coefficients, add the equations. Now, substitute −1 for x in either equation to find y. Now, substitute −2 for y in either equation to find x. The solution is (−1, −3). ANSWER: (−1, −3) The solution is (4, −2). ANSWER: (4, −2) 26. 2x + y = −5 x −y = 2 SOLUTION: Because y and −y have opposite coefficients, add the equations. Now, substitute −1 for x in either equation to find y. 27. 6x + y = 9 −6x + 3y = 15 SOLUTION: Because 6x and −6x have opposite coefficients, add the equations. Now, substitute 6 for y in either equation to find x. The solution is eSolutions Manual - Powered by Cognero The solution is (−1, −3). ANSWER: . Page 10 The solution is (−1, −3). ANSWER: ANSWER: Study Guide and Review - Chapter 6 (−1, −3) 27. 6x + y = 9 −6x + 3y = 15 28. x − 4y = 2 3x + 4y = 38 SOLUTION: Because 6x and −6x have opposite coefficients, add the equations. SOLUTION: Because −4y and 4y have opposite coefficients, add the equations. Now, substitute 6 for y in either equation to find x. Now, substitute 10 for x in either equation to find y. The solution is (10, 2). The solution is . ANSWER: 28. x − 4y = 2 3x + 4y = 38 SOLUTION: Because −4y and 4y have opposite coefficients, add the equations. ANSWER: (10, 2) 29. 2x + 2y = 4 2x − 8y = −46 SOLUTION: Because 2x and 2x have the same coefficients, you need to multiply equation 2 by -1 so they are additive inverses. Then add the equations. Now, substitute 10 for x in either equation to find y. Now, substitute 5 for y in either equation to find x. The solution is (10, 2). eSolutions Manual - Powered by Cognero ANSWER: (10, 2) Page 11 The solution is (−3, 5). The solution is (10, 2). The solution is (−3, 5). ANSWER: Study Guide and Review - Chapter 6 (10, 2) 29. 2x + 2y = 4 2x − 8y = −46 ANSWER: (−3, 5) 30. 3x + 2y = 8 x + 2y = 2 SOLUTION: Because 2x and 2x have the same coefficients, you need to multiply equation 2 by -1 so they are additive inverses. Then add the equations. SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add the equations. Now, substitute 5 for y in either equation to find x. Now, substitute 3 for x in either equation to find y. The solution is (−3, 5). The solution is ANSWER: (−3, 5) . ANSWER: 30. 3x + 2y = 8 x + 2y = 2 SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add the equations. 31. BASEBALL CARDS Cristiano bought 24 baseball cards for $50. One type cost $1 per card, and the other cost $3 per card. Define the variables, and write equations to find the number of each type of card he bought. Solve by using elimination. SOLUTION: Sample answer: Let f be the first type of card and let c be the second type of card. f + c = 24 (total number of cards) f + 3c = 50 (cost of the cards) Because f and f have the same coefficients, multiply equation 2 by –1 and then add the equations. eSolutions Manual - Powered by xCognero Now, substitute 3 for in either Page 12 equation to find y. ANSWER: ANSWER: Study Guide and Review - Chapter 6 31. BASEBALL CARDS Cristiano bought 24 baseball cards for $50. One type cost $1 per card, and the other cost $3 per card. Define the variables, and write equations to find the number of each type of card he bought. Solve by using elimination. SOLUTION: Sample answer: Let f be the first type of card and let c be the second type of card. f + c = 24 (total number of cards) f + 3c = 50 (cost of the cards) Because f and f have the same coefficients, multiply equation 2 by –1 and then add the equations. Sample answer: Let f be the number of the first type of card, and let c be the number of the second type of card; f + c = 24, f + 3c = 50; 11 $1 cards and 13 $3 cards. Use elimination to solve each system of equations. 32. x + y = 4 −2x + 3y = 7 SOLUTION: Notice that if you multiply the first equation by 2, the coefficients of the x-terms are additive inverses. Now, substitute 3 for y in either equation to find x. The solution is (1, 3). Now, substitute 13 for c in either equation to find f . The solution is (11, 13). Cristiano bought 11 $1 cards and 13 $3 cards. ANSWER: (1, 3) 33. x − y = −2 2x + 4y = 38 SOLUTION: Notice that if you multiply the first equation by 4, the coefficients of the y-terms are additive inverses. ANSWER: Sample answer: Let f be the number of the first type of card, and let c be the number of the second type of card; f + c = 24, f + 3c = 50; 11 $1 cards and 13 $3 cards. Now, substitute 5 for x in either equation to find y. Use elimination to solve each system of equations. 32. x + y = 4 −2x + 3y = 7 SOLUTION: Notice that if you multiply the first equation by 2, the coefficients of the x-terms are additive inverses. eSolutions Manual - Powered by Cognero The solution is (5, 7). ANSWER: (5, 7) 34. 3x + 4y = 1 5x + 2y = 11 Page 13 SOLUTION: Notice that if you multiply the second equation by −2, The solution is (5, 7). ANSWER: Study Guide and Review - Chapter 6 (5, 7) 34. 3x + 4y = 1 5x + 2y = 11 SOLUTION: Notice that if you multiply the second equation by −2, the coefficients of the y-terms are additive inverses. Now, substitute 3 for x in either equation to find y. The solution is (3, −2). ANSWER: (3, −2) 35. −9x + 3y = −3 3x − 2y = −4 SOLUTION: Notice that if you multiply the second equation by 3, the coefficients of the x-terms are additive inverses. Now, substitute 5 for y in either equation to find x. The solution is (2, 5). ANSWER: (2, 5) 36. 8x − 3y = −35 3x + 4y = 33 SOLUTION: Notice that if you multiply the first equation by 4 and the second equation by 3, the coefficients of the yterms are additive inverses. Now, substitute −1 for x in either equation to find y. The solution is (−1, 9). ANSWER: (−1, 9) 37. 2x + 9y = 3 5x + 4y = 26 SOLUTION: Notice that if you multiply the first equation by 5 and the second equation by −2, the coefficients of xterms are additive inverses. Now, substitute −1 for y in either equation to find x. The solution is (2, 5). ANSWER: (2, 5) 36. 8x − 3y = −35 eSolutions Manual - Powered by Cognero 3x + 4y = 33 SOLUTION: The solution is (6, −1). ANSWER: (6, −1) 38. −7x + 3y = 12 2x − 8y = −32 Page 14 The solution is (−1, 9). ANSWER: Study Guide and Review - Chapter 6 (−1, 9) 37. 2x + 9y = 3 5x + 4y = 26 The solution is (6, −1). ANSWER: (6, −1) 38. −7x + 3y = 12 2x − 8y = −32 SOLUTION: Notice that if you multiply the first equation by 5 and the second equation by −2, the coefficients of xterms are additive inverses. SOLUTION: Notice that if you multiply the first equation by 2 and the second equation by 7, the coefficients of the xterms are additive inverses. Now, substitute −1 for y in either equation to find x. The solution is (6, −1). ANSWER: (6, −1) ANSWER: (0, 4) 38. −7x + 3y = 12 2x − 8y = −32 39. 8x − 5y = 18 6x + 6y = −6 Now, substitute 4 for y in either equation to find x. The solution is (0, 4). SOLUTION: Notice that if you multiply the first equation by 2 and the second equation by 7, the coefficients of the xterms are additive inverses. SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the yterms are additive inverses. Now, substitute 4 for y in either equation to find x. Now, substitute 1 for x in either equation to find y. The solution is (0, 4). The solution is (1, −2). ANSWER: (0, 4) ANSWER: (1, −2) eSolutions Manual - Powered by Cognero 39. 8x − 5y = 18 6x + 6y = −6 Page 15 40. BAKE SALE On the first day, a total of 40 items were sold for $356. Define the variables, and write a The solution is (0, 4). The solution is (1, −2). ANSWER: Study Guide and Review - Chapter 6 (0, 4) 39. 8x − 5y = 18 6x + 6y = −6 SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the yterms are additive inverses. ANSWER: (1, −2) 40. BAKE SALE On the first day, a total of 40 items were sold for $356. Define the variables, and write a system of equations to find the number of cakes and pies sold. Solve by using elimination. Now, substitute 1 for x in either equation to find y. The solution is (1, −2). ANSWER: (1, −2) 40. BAKE SALE On the first day, a total of 40 items were sold for $356. Define the variables, and write a system of equations to find the number of cakes and pies sold. Solve by using elimination. SOLUTION: Let c represent the cakes and let p represent the pies. 8c + 10p = 356 c + p = 40 Notice that if you multiply the second equation by −8, the coefficients of the c-terms are additive inverses. Now, substitute 18 for p in either equation to find c. The solution is (22, 18). The Monarch Band Booster sold 22 cakes and 18 pies. SOLUTION: Let c represent the cakes and let p represent the pies. 8c + 10p = 356 c + p = 40 Notice that if you multiply the second equation by −8, the coefficients of the c-terms are additive inverses. eSolutions Manual - Powered by Cognero ANSWER: Sample answer: Let c represent the number of the cakes, and let p represent the number of pies; 8c + 10p = 356, p + c = 40; 22 cakes, 18 pies Determine the best method to solve each system of equations. Then solve the system. 41. y = x − 8 y = −3x SOLUTION: Because both equations are solved for one of the variables, substitution is the best method. Page 16 Substitute −3x for y in the first equation. ANSWER: Sample answer: Let c represent the number of the Study Guide - Chapter 6 of pies; 8c + cakes, andand let pReview represent the number 10p = 356, p + c = 40; 22 cakes, 18 pies Determine the best method to solve each system of equations. Then solve the system. 41. y = x − 8 y = −3x SOLUTION: Because both equations are solved for one of the variables, substitution is the best method. Substitute −3x for y in the first equation. Substitute 2 for x in either equation to find y. ANSWER: Subs.; (2, −6) 42. y = −x y = 2x SOLUTION: Because both equations are solved for one of the variables, substitution is the best method. Substitute 2x for y in the first equation. Substitute 0 for x in either equation to find y. The solution is (0, 0). The solution is (2, −6). ANSWER: Subs.; (2, −6) 42. y = −x y = 2x SOLUTION: Because both equations are solved for one of the variables, substitution is the best method. Substitute 2x for y in the first equation. ANSWER: Subs.; (0, 0) 43. x + 3y = 12 x = −6y SOLUTION: Because one of the equations is solved for one of the variables, substitution is the best method. Substitute −6y for x in the first equation. Substitute −4 for y in either equation to find x. Substitute 0 for x in either equation to find y. The solution is (24, −4). The solution is (0, 0). ANSWER: Subs.; (0, 0) eSolutions Manual - Powered by Cognero ANSWER: Subs.; (24, −4) 44. x + y = 10 x − y = 18 Page 17 The solution is (0, 0). ANSWER: Subs.; Study Guide and Review - Chapter 6 (0, 0) 43. x + 3y = 12 x = −6y SOLUTION: Because one of the equations is solved for one of the variables, substitution is the best method. Substitute −6y for x in the first equation. Substitute −4 for y in either equation to find x. The solution is (24, −4). ANSWER: Subs.; (24, −4) 44. x + y = 10 x − y = 18 SOLUTION: Because y and −y have opposite coefficients, elimination using addition is the best method. Now, substitute 14 for x in either equation to find y. The solution is (24, −4). The solution is (14, −4). ANSWER: Subs.; (24, −4) ANSWER: Elim (+); (14, −4) 44. x + y = 10 x − y = 18 SOLUTION: Because y and −y have opposite coefficients, elimination using addition is the best method. 45. 3x + 2y = −4 5x + 2y = −8 SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by –1. Now, substitute 14 for x in either equation to find y. Now, substitute −2 for x in either equation to find y. The solution is (14, −4). ANSWER: Elim (+); (14, −4) eSolutions Manual - Powered by Cognero 45. 3x + 2y = −4 5x + 2y = −8 Page 18 The solution is (14, −4). ANSWER: Elim (+); Study Guide and Review - Chapter 6 (14, −4) 45. 3x + 2y = −4 5x + 2y = −8 SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by –1. The solution is (−2, 1). ANSWER: Elim (–); (−2, 1) 46. 6x + 5y = 9 −2x + 4y = 14 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the second equation by 3, the coefficients of the x-terms are additive inverses. Now, substitute 3 for y in either equation to find x. Now, substitute −2 for x in either equation to find y. The solution is (−1, 3). ANSWER: Elim (×); (−1, 3) The solution is (−2, 1). ANSWER: Elim (–); (−2, 1) 46. 6x + 5y = 9 −2x + 4y = 14 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the second equation by 3, the coefficients of the x-terms are additive inverses. 47. 3x + 4y = 26 2x + 3y = 19 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 2 and the second equation by −3, the coefficients of the xterms are additive inverses. Now, substitute 5 for y in either equation to find x. Now, substitute 3 for y in either equation to find x. eSolutions Manual - Powered by Cognero The solution is (2, 5). ANSWER: Page 19 The solution is (−1, 3). ANSWER: Study Guide and Review - Chapter 6 Elim (×); (−1, 3) 47. 3x + 4y = 26 2x + 3y = 19 The solution is (2, 5). ANSWER: Elim (×); (2, 5) 48. 11x − 6y = 3 5x − 8y = −25 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 2 and the second equation by −3, the coefficients of the xterms are additive inverses. SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the yterms are additive inverses. Now, substitute 3 for x in either equation to find y. Now, substitute 5 for y in either equation to find x. The solution is (2, 5). The solution is (3, 5). ANSWER: Elim (×); (2, 5) ANSWER: Elim (×); (3, 5) 48. 11x − 6y = 3 5x − 8y = −25 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the yterms are additive inverses. Now, substitute 3 for x in either equation to find y. eSolutions Manual - Powered by Cognero The solution is (3, 5). 49. COINS Tionna has 25 coins in her piggy bank with a value of $4. The coins are either dimes or quarters. Define the variables, and write a system of equations to determine the number of dimes and quarters. Then solve the system using the best method for the situation. SOLUTION: Sample answer: Let d represent the number of dimes and let q represent the number of quarters. Use the fact that the value of a dime is $0.10 and the value of a quarter is $0.25 to write the equation for the value. d + q = 25 0.10d + 0.25q = 4 Because the coefficients of d and q in the first equation are 1, the best method is substitution. Solve the first equation for q. q = −d + 25 Substitute −d + 25 for q in the second equation. Page 20 The solution is (3, 5). ANSWER: Study Guide and Review - Chapter 6 Elim (×); (3, 5) 49. COINS Tionna has 25 coins in her piggy bank with a value of $4. The coins are either dimes or quarters. Define the variables, and write a system of equations to determine the number of dimes and quarters. Then solve the system using the best method for the situation. SOLUTION: Sample answer: Let d represent the number of dimes and let q represent the number of quarters. Use the fact that the value of a dime is $0.10 and the value of a quarter is $0.25 to write the equation for the value. d + q = 25 0.10d + 0.25q = 4 Because the coefficients of d and q in the first equation are 1, the best method is substitution. Solve the first equation for q. q = −d + 25 Substitute −d + 25 for q in the second equation. ANSWER: Sample answer: Let d represent the number of dimes and let q represent the number of quarters; d + q = 25, 0.10d + 0.25q = 4; 15 dimes, 10 quarters 50. FAIR At a county fair, the cost for 4 slices of pizza and 2 orders of French fries is $21.00. The cost of 2 slices of pizza and 3 orders of French fries is $16.50. To find out how much a single slice of pizza and an order of French fries costs, define the variables and write a system of equations to represent the situation. Determine the best method to solve the system of equations. Then solve the system. SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the second equation by -2, the coefficients of the p -terms are additive inverses. Now, substitute 3 for f in either equation to find p . Substitute 15 for d in either equation to find q. pizza: $3.75; French fries: $3 The solution is (15, 10). So, Tionna has 15 dimes and 10 quarters in her piggy bank. ANSWER: Sample answer: Let d represent the number of dimes and let q represent the number of quarters; d + q = 25, 0.10d + 0.25q = 4; 15 dimes, 10 quarters 50. FAIR At a county fair, the cost for 4 slices of pizza and 2 orders of French fries is $21.00. The cost of 2 slices of pizza and 3 orders of French fries is $16.50. To find out how much a single slice of pizza and an order of French fries costs, define the variables and write a system of equations to represent the situation. Determine the best eSolutions Manual - Powered by Cognero method to solve the system of equations. Then solve the system. ANSWER: Let p represent the cost of a slice of pizza and t represent the cost of an order of French fries; 4p + 2f = 21, 2p + 3f = 16.5; Sample answer: elimination; pizza $3.75; French fries $3. Solve each system of inequalities by graphing. 51. x > 3 y <x+2 SOLUTION: Graph each inequality. The graph of x > 3 is dashed and is not included in the graph of the solution. Page 21 y <x+2 SOLUTION: Graph each inequality. Study Guide and Review - Chapter 6 The graph of x > 3 is dashed and is not included in the graph of the solution. ANSWER: The graph of y < x + 2 is also dashed and is not included in the graph of the solution. 52. y ≤ 5 y >x−4 SOLUTION: Graph each inequality. The graph of y ≤ 5 is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 3 and y < x + 2. Overlay the graphs and locate the green region. This is the intersection. The graph of y > x − 4 is dashed and is not included in the graph of the solution. The solution region is shaded in the graph below. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≤ 5 and y > x − 4. Overlay the graphs and locate the green region. This is the intersection. ANSWER: eSolutions Manual - Powered by Cognero Page 22 The solution region is shaded in the graph below. The solution of the system is the set of ordered pairs in Guide the intersection of the- Chapter graphs of 6y ≤ 5 and y > x − Study and Review 4. Overlay the graphs and locate the green region. This is the intersection. The graph of y ≥ −2x + 4 is solid and is included in the graph of the solution. The solution region is shaded in the graph below. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x − 1 and y ≥ −2x + 4. Overlay the graphs and locate the green region. This is the intersection. ANSWER: The solution region is shaded in the graph below. 53. y < 3x − 1 y ≥ −2x + 4 SOLUTION: Graph each inequality. The graph of y < 3x − 1 is dashed and is not included in the graph of the solution. ANSWER: 54. y ≤ −x − 3 y ≥ 3x − 2 The Manual graph -ofPowered y ≥ −2x 4 is solid and is included in eSolutions by + Cognero the graph of the solution. SOLUTION: Page 23 Graph each inequality. The graph of y ≤ −x − 3 is solid and is included in Study 54. y ≤Guide −x − 3and Review - Chapter 6 y ≥ 3x − 2 ANSWER: SOLUTION: Graph each inequality. The graph of y ≤ −x − 3 is solid and is included in the graph of the solution. The graph of y ≥ 3x − 2 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≤ −x − 3 and y ≥ 3x − 2. Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below. ANSWER: eSolutions Manual - Powered by Cognero Page 24