Question

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

Emilia, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $13 plus $7 per kilogram. The second charges $10 plus $8 per kilogram. Emilia determines that, given her package's weight, the two courier services are equivalent in terms of cost. How much will it cost? What is the weight?


The two couriers both cost $
at a package weight of
kilograms.

Answer 1

To model the situation, we can create a system of equations based on the costs of the two couriers.

Let \( x \) be the weight of the package in kilograms.

The cost for the first courier can be expressed as:
\[
C_1 = 13 + 7x
\]
The cost for the second courier can be expressed as:
\[
C_2 = 10 + 8x
\]

Since both courier services are equivalent in terms of cost, we set \( C_1 \) equal to \( C_2 \):

\[
13 + 7x = 10 + 8x
\]

Now, we will solve for \( x \).

1. Subtract \( 7x \) from both sides:
\[
13 = 10 + 8x - 7x
\]
\[
13 = 10 + x
\]

2. Subtract 10 from both sides:
\[
13 - 10 = x
\]
\[
x = 3
\]

Now that we have the weight of the package, we can find the cost by substituting \( x \) back into either courier's cost equation. Let's use the first courier's cost equation:

\[
C_1 = 13 + 7(3)
\]
\[
C_1 = 13 + 21
\]
\[
C_1 = 34
\]

So, both couriers cost the same amount.

The two couriers both cost $34 at a package weight of 3 kilograms.

Filling in the blanks:

The two couriers both cost **$34** at a package weight of **3** kilograms.