-2x + 2y = 2 4x –4y= 4A. No solutionB. One solution: x = y = ___C. Infinitely many solutionsBONUS: A customer at a concession stand bought 2 boxes of popcorn and 3 drinks for $12. Another customerbought 3 boxes of popcorn and 5 drinks for $19. How much does a box of popcorn cost? How much does a drink cost?

See answers below

Step-by-step explanation:

1) Given the simultaneous equation;

-2x + 2y = 2 1  * 2

4x –4y= 4...2 * 1

-4x + 4y = 4 1

4x –4y= 4...2

Add both equations

4y-4y = 4-4

0y = 0

y = 0/0

Let x = k

Substitute x = k into 1

From 1: -2x + 2y = 2

-x+y = 1

-k + y = 1

y = 1+k

Hence the solution to the system of equation (x, y) is (k, 1 + k) where k is any integer.

The presence of k shows that the equation has many solutions since k can take any value

2) Let a box of popcorn be x

Let the cost of a drink be y

If a customer at a concession stand bought 2 boxes of popcorn and 3 drinks for $12, then;

2x + 3y = 12 ... 1

Also, if another customer bought 3 boxes of popcorn and 5 drinks for $19, then;

3x + 5y = 19 ...2

Solve 1 and 2 simultaneously

2x + 3y = 12 ... 1  * 3

3x + 5y = 19 ...2  * 2

6x + 9y = 36  

6x + 10y = 38

Subtract

9y - 10y = 36 - 38

-y = -2

y = 2

Recall that 2x + 3y = 12

2x + 3(2) = 12

2x + 6 = 12

2x = 12-6

2x = 6

x = 6/2

x= 3

Hence a box of popcorn cost $3 and a drink cost $2

7. d. 64

8. i. 32

9. c. 125

10. j. none of the above

11. d. 81

12. i. 216

13. e. none of the above

14. g. 240

15. e. none of the above

16. g. 32

17.   e. none of the above