Question

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

The postal service offers flat-rate shipping for priority mail in special boxes. Today, Toby shipped 5 small boxes and 2 large boxes, which cost him $50 to ship. Meanwhile, Clara shipped 1 small box and 2 large boxes, and paid $34. How much does it cost to ship these two sizes of box?

Shipping costs $
for a small box and $
for a large box.

Answer 1

Let's denote the cost to ship a small box as \( x \) and the cost to ship a large box as \( y \).

Based on the information provided, we can set up the following system of equations:

1. Toby's shipping:
\[
5x + 2y = 50
\]

2. Clara's shipping:
\[
1x + 2y = 34
\]

Now, we have the system of equations:
\[
\begin{align*}
1. & \quad 5x + 2y = 50 \quad \text{(Equation 1)} \\
2. & \quad 1x + 2y = 34 \quad \text{(Equation 2)}
\end{align*}
\]

To solve this system, we can use the elimination method. First, let's subtract Equation 2 from Equation 1:

\[
(5x + 2y) - (1x + 2y) = 50 - 34
\]

This simplifies to:

\[
4x = 16
\]

Next, we can solve for \( x \):

\[
x = \frac{16}{4} = 4
\]

Now that we have \( x \), we can substitute it back into Equation 2 to solve for \( y \):

\[
1(4) + 2y = 34
\]

This simplifies to:

\[
4 + 2y = 34
\]

Subtract 4 from both sides:

\[
2y = 30
\]

Now solve for \( y \):

\[
y = \frac{30}{2} = 15
\]

Thus, the costs are:
- The cost of a small box (\( x \)) is \( 4 \) dollars.
- The cost of a large box (\( y \)) is \( 15 \) dollars.

Now we can fill in the blanks:

Shipping costs **$4** for a small box and **$15** for a large box.