Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent.

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1 Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 2x 1 y = 2x + 3 The lines y = 2x 1 and y = 2x + 3 intersect at exactly one point which means this system has exactly one solution. So, the system is consistent and independent. y = 2x + 3 y = 2x 3 The lines y = 2x + 3 and y = 2x 3 never intersect which means this system has no solution. So, the system is inconsistent. esolutions Manual - Powered by Cognero Page 1

2 Graph each system and determine the number of solutions that it has. If it has one solution, name it. y = 2x 3 y = x + 4 y = 2x 3 y = x + 4 The graphs appear to intersect at the point (7, 11). You can check this by substituting 7 for x and 11 for y. The solution is (7, 11). esolutions Manual - Powered by Cognero Page 2

3 x + y = 6 x y = 4 To graph the system, write both equations in slope-intercept form. y = x + 6 y = x 4 The graphs appear to intersect at the point (5, 1). You can check this by substituting 5 for x and 1 for y. The solution is (5, 1). x + y = 8 3x + 3y = 24 To graph the system, write both equations in slope-intercept form. y = x + 8 y = x + 8 When written in slope-intercept form, you can see that the equations represent the same line. There are an infinite number of solutions. esolutions Manual - Powered by Cognero Page 3

4 x 4y = 6 y = 1 To graph the system, write both equations in slope-intercept form. Graph and y = 1. The graphs appear to intersect at the point (10, 1). You can check this by substituting 10 for x and 1 for y. The solution is (10, 1) = 12 esolutions Manual - Powered by Cognero Page 4

5 3x + 2y = 12 3x + 2y = 6 To graph the system, write both equations in slope-intercept form. Graph and. The lines are parallel. So, there is no solution. esolutions Manual - Powered by Cognero Page 5

6 2x + y = 4 5x + 3y = 6 To graph the system, write both equations in slope-intercept form. Equation 1: Equation 2: Graph y = 2x 4 and. The graphs appear to intersect at the point (6, 8). You can check this by substituting 6 for x and 8 for y. The solution is (6, 8). esolutions Manual - Powered by Cognero Page 6

7 Use substitution to solve each system of equations. y = x + 4 2x + y = 16 y = x + 4 2x + y = 16 Substitute x + 4 for y in the second equation. Use the solution for x and either equation to find the value for y. The solution is (4, 8). y = 2x 3 x + y = 9 y = 2x 3 x + y = 9 Substitute 2x 3 for y in the second equation. Use the solution for x and either equation to find the value for y. The solution is (12, 21). esolutions Manual - Powered by Cognero Page 7

8 x + y = 6 x y = 8 x + y = 6 x y = 8 Solve the first equation for x. Substitute 6 y for x in the second equation. Use the solution for y and either equation to find the value for x. The solution is (7, 1). y = 4x 6x y = 30 y = 4x 6x y = 30 Substitute 4x for y in the second equation. Use the solution for x and either equation to find the value for y. The solution is (3, 12). esolutions Manual - Powered by Cognero Page 8

9 The cost of two meals at a restaurant is shown in the table. Define variables to represent the cost of a taco and the cost of a burrito. Write a system of equations to find the cost of a single taco and a single burrito. Solve the systems of equations, and explain what the solution means. How much would a customer pay for 2 tacos and 2 burritos? Let t = the cost of a taco and b = the cost of a burrito. The cost of a meal with 3 tacos and 2 burritos is $7.40. So, 3t + 2b = 7.4. The cost of a meal with 4 tacos and 1 burrito is $6.45. So, 4t + b = c. 3t + 2b = 7.4 4t + b = 6.45 Solve the second equation for b. Substitute 4t for b in the first equation. Use the solution for t in either equation to find the value of b. The cost of a single taco is $1.10 and the cost of a single burrito is $2.05. d. Substitute these values into the equation to find how much a customer pays for 2 tacos and 2 burritos. esolutions Manual - Powered by Cognero Page 9

10 A customer would pay $6.30 for 2 tacos and 2 burritos. The cost of two groups going to an amusement park is shown in the table. Define variables to represent the cost of an adult ticket and the cost of a child ticket. Write a system of equations to find the cost of an adult and child admission. Solve the system of equations, and explain what the solution means. d. Let a = cost of an adult ticket and c = the cost of a child ticket. The cost of a group with 4 adults and 2 children is $184. So, 4a + 2c = 184. The cost of a group with 4 adults and 3 children is $200. So, 4a + 3c = 200. Notice that the coefficients of the a terms are the same, so subtract the equations. Use the solution for c in either equation to find the value of a. The cost of an adults ticket is $38, and the cost of a childs ticket is $16. Substitute these values into the equation to find the total cost of admission. A group of 3 adults and 5 children visiting the amusement park will be charged $194 for admission. Angelina spends $16 for 12 pieces of candy to take to a meeting. Each chocolate bar esolutions Manual - Powered by Cognero Page 10

11 Angelina spends $16 for 12 pieces of candy to take to a meeting. Each chocolate bar costs $2, and each lollipop costs $1. Determine how many of each she bought. 6 chocolate bars, 6 lollipops 4 chocolate bars, 8 lollipops 7 chocolate bars, 5 lollipops 3 chocolate bars, 9 lollipops Let c = the number of chocolate bars and = the number of lollipops. Angelina buys 12 pieces of candy. So, c + = 12. She spends $16. So, 2c + 1 = 16. Since both equations contain, use elimination by subtraction. Substitute 4 for c in either equation to solve for. So, B is the correct choice. esolutions Manual - Powered by Cognero Page 11

12 Use elimination to solve each system of equations. x + y = 9 x y = 3 Because y and y have opposite coefficients, add the equations. Now, substitute 3 for x in either equation to find the value of y. The solution is (3, 6). x + 3y = 11 x + 7y = 19 Because x and x have the same coefficients, subtract the equations. Now, substitute 2 for y in either equation to find the value of x. The solution is (5, 2). esolutions Manual - Powered by Cognero Page 12

13 9x 24y = 6 3x + 4y = 10 Multiply each term in the second equation by 3 to eliminate the x coefficient. Because 9x and 9x have opposite coefficients, add the equations. Now, substitute 1 for y in either equation to find the value of y. The solution is (2, 1). 5x + 2y = 11 5x 7y = 1 Because 5x and 5x have opposite coefficients, add the equations. Now, substitute 2 for y in either equation to find the value of x. The solution is (3, 2). The Blue Mountain High School Drama Club is selling tickets to their spring musical. esolutions Manual - Powered by Cognero Page 13

14 The Blue Mountain High School Drama Club is selling tickets to their spring musical. Adult tickets are $4 and student tickets are $1. A total of 285 tickets are sold for $765. How many of each type of ticket are sold? 145 adult, 140 student 120 adult, 165 student 180 adult, 105 student 160 adult, 125 student Let a = the number of adult tickets sold and s = the number of student tickets sold. So, a + s = 285 and 4a + 1s = 765. Solve the first equation for s. Substitute 285 a for s in the second equation. Use the solution for a in either equation to find the value of s. So, J is the correct choice. esolutions Manual - Powered by Cognero Page 14

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