Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Question

Kenneth and Nellie just had business cards made. Kenneth's printing company charged a one-time setup fee of $17 and then $13 per box of cards. Nellie, meanwhile, ordered hers online. They cost $12 per box. There was no setup fee, but she had to pay $20 to have her order shipped to her house. By coincidence, Kenneth and Nellie ended up spending the same amount on their business cards. How many boxes did each buy? How much did each spend?

Answer 1

A system of equations is two or more systems working together. In this example there is Kenneth's and Nellie's cost for business cards. Modeling each, you would have:

Kenneth's Cost = 13x + 17 ($13 per box plus $17 set-up charge)

Nellie's Cost = 12x + 20 ($12 per box plus $20 to ship order)

The question is how many boxes did each purchase in order for them to each spend the same amount. To solve for this, set each modeled cost equal to the other.

13x + 17 = 12x + 20

-12x -12x.

1x + 17 = 20

-17. -17.

1x = 3

so, when they each order 3 boxes, their cost is the same. To check this, plug 3 back into each model for x.

13(3) + 17 = 56

12(3) + 20 = 56