3-1 Solving Systems of Equations. Solve each system of equations by using a table. 2. SOLUTION: Write each equation in slope-intercept form.

Transcription

1 Solve each system of equations by using a table. 2. Write each equation in slope-intercept form. Also: Make a table to find a solution that satisfies both equations. Therefore, the solution of the system is (2, 7). Solve each system of equations by graphing. 4. Graph the equations y = x 9 and 3y = 5x + 5. The lines intersect at the point ( 4, 5). So, the solution of the system is ( 4, 5). esolutions Manual - Powered by Cognero Page 1

2 6. Graph the equations 3y = 4x + 11 and 2x + 3y = The lines intersect at the point ( 2, 1). So, the solution of the system is ( 2, 1). Graph the equations 8x 4y = 50 and x + 4y = 2. The lines intersect at the point (6, 2). So, the solution of the system is (6, 2). esolutions Manual - Powered by Cognero Page 2

3 10. Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. The lines are parallel. They do not intersect and there is no solution. So, the system is inconsistent. 12. The lines intersect at one point, so there is one solution. The system is consistent and independent. esolutions Manual - Powered by Cognero Page 3

4 14. Solve each system of equations by using substitution. Substitute 2x 10 for y in the equation and solve for x. Substitute 3 for x into either original equation and solve for y. The solution is (3, 4). 16. Solve the equation for a. Substitute 3b 22 for a in the equation and solve for b. Substitute 6 for b into either original equation and solve for a. The solution is ( 4, 6). esolutions Manual - Powered by Cognero Page 4

5 18. Solve the equation for c. Substitute for c in the equation and solve for d. Substitute 2 for d into either original equation and solve for c. The solution is ( 3, 2). esolutions Manual - Powered by Cognero Page 5

6 20. Solve each system of equations by using elimination. Add the equations to eliminate y. Substitute 8 for x into either original equation and solve for y. The solution is (8, 1). 22. The coefficients of b-variables are 3 and 2 and their least common multiple is 6, so multiply each equation by the value that will make the b-coefficient 6. Substitute 1 for a into either original equation and solve for b. The solution is ( 1, 1). esolutions Manual - Powered by Cognero Page 6

7 24. The coefficients of y-variables are 4 and 5 and their least common multiple is 20, so multiply each equation by the value that will make the y-coefficient 20. Substitute 1 for x into either original equation and solve for y. 26. The solution is (1, 6). Solve each system of equations by using a table. y = 5x + 3 y = x 9 Make a table to find a solution that satisfies both equations. Therefore, the solution of the system is ( 3, 12). esolutions Manual - Powered by Cognero Page 7

8 28. Write each equation in slope-intercept form. Also: Make a table to find a solution that satisfies both equations. 30. Therefore, the solution of the system is (4, 3). Solve each system of equations by graphing. Graph the equations 3x + 2y = 6 and 5x + 10y = 30. The lines intersect at the point (6, 6). So, the solution of the system is (6, 6). esolutions Manual - Powered by Cognero Page 8

9 32. Graph the equations 6x 5y = 17 and 6x + 2y = 31. The lines intersect at the point (4.5, 2). So, the solution of the system is (4.5, 2). 34. Graph the equations y 3x = 29 and 9x 6y = 102. The lines intersect at the point (8, 5). So, the solution of the system is (8, 5). esolutions Manual - Powered by Cognero Page 9

10 36. CCSS MODELING Jerilyn has a $10 coupon and a 15% discount coupon for her favorite store. The store has a policy that only one coupon may be used per purchase. When is it best for Jerilyn to use the $10 coupon, and when is it best for her to use the 15% discount coupon? Let $x be the total cost (excluding discounts) of the items that Jerilyn purchased. If she uses the $10 coupon, the cost will be $(x 10). If she uses the 15% coupon, the cost will be $0.85x. Assume: 38. So, when the total cost of the items purchased is less than $66.67, it is best to use the $10 coupon. The 15% discount is best when the purchase is more than $ Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Graph the equations y = 2x 1 and y = 2x + 6. The lines are parallel. They do not intersect and there is no solution. So, the system is inconsistent. esolutions Manual - Powered by Cognero Page 10

11 40. Graph the equations x 6y = 12 and 3x + 18y = 14. The lines intersect at one point, so there is one solution. The system is consistent and independent. 42. Graph the equations 8y 3x = 15 and 16y + 6x = 30. Because the equations are equivalent, their graphs are the same line. The system is consistent and dependent as it has an infinite number of solutions. esolutions Manual - Powered by Cognero Page 11

12 44. Solve each system of equations by using substitution. Solve the equation for x. Substitute y for x in the equation and solve for y. Substitute 4 for y into either original equation and solve for x. The solution is ( 2, 4). esolutions Manual - Powered by Cognero Page 12

13 46. Solve the equation for b. Substitute 16 3a for b in the equation and solve for a. Substitute 6 for a into either original equation and solve for b. The solution is ( 6, 2). 48. Solve the equation for g. Substitute for g in the equation and solve for f. Because 0 = 96 is not true, this system has no solution. esolutions Manual - Powered by Cognero Page 13

14 50. Solve each system of equations by using elimination. Multiply the equation by 4. Add the equations to eliminate one variable. Substitute 3 for x into either original equation and solve for y. The solution is (3, 3). esolutions Manual - Powered by Cognero Page 14

15 52. Subtract the equations to eliminate one variable. Substitute 5 for j into either original equation and solve for k. The solution is ( 5, 4). 54. The coefficients of b-variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value that will make the b-coefficient 10. Substitute 5 for a into either original equation and solve for b. The solution is ( 5, 3). esolutions Manual - Powered by Cognero Page 15

16 56. The coefficients of f -variables are 5 and 9 and their least common multiple is 45, so multiply each equation by the value that will make the f -coefficient 45. Substitute 2 for d into either original equation and solve for f. The solution is ( 2, 4). 58. Rewrite the equation in the standard form Multiply the equation by 5. Add the equations to eliminate one variable. Because 0 = 0 is true, the system has infinite solutions. esolutions Manual - Powered by Cognero Page 16

17 60. Use a graphing calculator to solve each system of equations. Round the coordinates of the intersection to the nearest hundredth. Write each equation in the form y = mx + b. Enter as Y1 and as Y2. Then graph the lines. KEYSTROKES: Y = ( ) 2 9 ENTER ( ) 5 4 ENTER ZOOM 3 ENTER ENTER Find the intersection of the lines. KEYSTROKES: 2nd [CALC] 5 ENTER ENTER ENTER The coordinates of the intersection to the nearest hundredth are. esolutions Manual - Powered by Cognero Page 17

18 62. Solve each system of equations. Multiply the equation by 4. Add the equations to eliminate one variable. Because 0 = 0 is true, the system has infinite solutions. 64. The coefficients of t-variables are 10 and 4 and their least common multiple is 20, so multiply each equation by the value that will make the t-coefficient 20. Substitute 3 for v into either original equation and solve for t. The solution is (2.5, 3). esolutions Manual - Powered by Cognero Page 18

19 66. Solve the equation for z. Substitute for z in the equation and solve for y. Because 0 = 0 is true, the system has infinite solutions. esolutions Manual - Powered by Cognero Page 19

20 68. ROWING Allison can row a boat 1 mile upstream (against the current) in 24 minutes. She can row the same distance downstream in 13 minutes. Assume that both the rowing speed and the speed of the current are constant. a. Find the speed at which Allison is rowing and the speed of the current. b. If Allison plans to meet her friends 3 miles upstream one hour from now, will she be on time? Explain. a. Let x be the rowing speed. Let y be the speed of the current. Allison s speed in upstream: Allison s speed in downstream: The system of equations that represents this situation is Add the equations to eliminate one variable. Substitute 3.56 for x into either original equation and solve for y. The speed of the rowing boat is 3.56 mph and the speed of the current is 1.06 mph. b. No, Allison will be late. To row 3 miles upstream she needs 72 minutes. So, = 12 minutes. She will be 12 minutes late. esolutions Manual - Powered by Cognero Page 20

21 70. JOBS Levi has a job offer in which he will receive $800 per month plus a commission of 2% of the total price of cars he sells. At his current job, he receives $1200 per month plus a commission of 1.5% of his total sales. How much must he sell per month to make the new job a better deal? Let x be the total price of cars Levi sells. To make the new job a better deal, he has to sell at least $80,000 worth of cars. esolutions Manual - Powered by Cognero Page 21

22 GEOMETRY Find the point at which the diagonals of the quadrilaterals intersect. 72. The quadrilateral has the vertices A(2, 2), B(0, 8), C(10, 5), and D(8, 2). The equation of the lines Substitute for y in the equation and solve for x. Substitute 6 for x into either original equation and solve for y. The diagonals of the quadrilateral intersect at (6, 3.5). esolutions Manual - Powered by Cognero Page 22

23 74. ELECTIONS In the election for student council, Candidate A received 55% of the total votes, while Candidate B received 1541 votes. If Candidate C received 40% of the votes that Candidate A received, how many total votes were cast? Let x be the total number of votes. The equation that represents the situation is Solve for x. The total number of votes is CCSS CRITIQUE Gloria and Syreeta are solving the system and Is either of them correct? Explain your reasoning. Sample answer: Gloria is correct. Syreeta subtracted 26 from 17 instead of 17 from 26 and got 3x = 9 instead of 3x = 9. She proceeded to get a value of 11 for y. She would have found her error if she substituted the solution into the original equations. esolutions Manual - Powered by Cognero Page 23

24 78. REASONING If a is consistent and dependent with b, b is inconsistent with c, and c is consistent and independent with d, then a will sometimes, always, or never be consistent and independent with d. Explain your reasoning. Sample answer: Always; a and b are the same line. b is parallel to c, so a is also parallel to c. Since c and d are consistent and independent, then c is not parallel to d and, thus, intersects d. Since a and c are parallel, then a cannot be parallel to d, so, a must intersect d and must be consistent and independent with d. 80. WRITING IN MATH Why is substitution sometimes more helpful than elimination, and vice versa? Sample answer: It is more helpful to use substitution when one of the variables has a coefficient of 1 or if a coefficient can be reduced to 1 without turning other coefficients into fractions. Otherwise, elimination is more helpful because it will avoid the use of fractions in solving the system. 82. ACT/SAT Which of the following best describes the graph of the equations? A The lines are parallel. B The lines are perpendicular. C The lines have the same x-intercept. D The lines have the same y-intercept. E The lines are the same. Write each equation in slope-intercept form. Also: The slopes are equal. So the lines are parallel. The correct choice is A. esolutions Manual - Powered by Cognero Page 24

25 84. Move-A-Lot Rentals will rent a moving truck for $100 plus $0.10 for every mile it is driven. Which equation can be used to find C, the cost of renting a moving truck and driving it for m miles? A C = 0.1(100 + m) B C = m C C = 100m D C = 100(m + 0.1) Let m be the number of miles the truck is driven. So: C = m The correct choice is B. Write an equation for each function. 86. The graph is a translation of the graph of y = x 2. When a constant k is added to or subtracted from a parent function, the result f (x) ± k is a translation of the graph up or down. Here, the graph of the parent function is moved 6 units up. So, 6 should be added to the parent function. So, the equation of the graph is. 88. The graph is a transformation of the graph of the parent function y = x 2. When a constant h is added to or subtracted from x before evaluating a parent function, the result, f (x ± h), is a translation left or right. When a constant k is added to or subtracted from a parent function, the result f (x) ± k is a translation of the graph up or down. A reflection flips a figure over a line called line of reflection. The reflection f (x) reflects the graph of f (x) across the x-axis and the reflection f ( x) reflects the graph of f (x) across the y-axis. Here, the graph of the parent function is moved 4 units right and reflected across the x-axis. So, the equation of the graph is. esolutions Manual - Powered by Cognero Page 25

26 Solve each equation. Check your solution n = 6 Substitute n = 8 in the original equation and check. So, the solution is n = x + 9x 6x + 4x = 20 Substitute x = 2.5 in the original equation and check. So, the solution is x = 2.5. esolutions Manual - Powered by Cognero Page 26

27 94. 7(p + 7) + 3(p 4) = 17 Substitute p = 11 in the original equation and check. So, the solution is p = 11. Determine whether the given point satisfies each inequality. 96. Substitute 1 for x and 1 for y in the inequality. The point (1, 1) satisfies the inequality esolutions Manual - Powered by Cognero Page 27