Question

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

Preston and Antonio just had business cards made. Preston's printing company charged a one-time setup fee of $20 and then $10 per box of cards. Antonio, meanwhile, ordered his online. They cost $11 per box. There was no setup fee, but he had to pay $19 to have his order shipped to his house. By coincidence, Preston and Antonio ended up spending the same amount on their business cards. How many boxes did each buy? How much did each spend?

Answer 1

Let \( x \) represent the number of boxes of business cards purchased by both Preston and Antonio.

**Equations:**

1. For Preston:
- Total cost = Setup fee + Cost per box * Number of boxes
- Cost for Preston: \( P = 20 + 10x \)

2. For Antonio:
- Total cost = Shipping fee + Cost per box * Number of boxes
- Cost for Antonio: \( A = 19 + 11x \)

Since Preston and Antonio spent the same amount, we can set their cost equations equal to each other:

\[
20 + 10x = 19 + 11x
\]

**Solving the equation:**

First, we will simplify the equation by isolating \( x \):

1. Subtract \( 10x \) from both sides:
\[
20 = 19 + 11x - 10x
\]
\[
20 = 19 + x
\]

2. Subtract 19 from both sides:
\[
20 - 19 = x
\]
\[
1 = x
\]

So, \( x = 1 \).

**Calculating the costs:**

Now, we can find the total cost for each person:

- Preston's total cost:
\[
P = 20 + 10(1) = 20 + 10 = 30
\]

- Antonio's total cost:
\[
A = 19 + 11(1) = 19 + 11 = 30
\]

Thus, both spent $30.

**Filling in the blanks:**

Preston and Antonio each bought **1** box of business cards. Each spent **$30**.