Question

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

A boy scout troop is selling Christmas trees at a local tree lot. In the morning, they sold 1 Douglas Fir tree and 7 Noble Fir trees, earning a total of $550. In the afternoon, they sold 1 Douglas Fir tree and 18 Noble Fir trees, earning a total of $1,342. How much does each type of tree cost?

A Douglas Fir costs $
and a Noble Fir costs $
.

Answer 1

Let's define the variables:

- Let \( d \) be the cost of one Douglas Fir tree.
- Let \( n \) be the cost of one Noble Fir tree.

According to the problem, we have the following system of equations from the sales data:

1. In the morning, they sold 1 Douglas Fir and 7 Noble Firs for a total of $550:
\[
d + 7n = 550 \quad \text{(1)}
\]

2. In the afternoon, they sold 1 Douglas Fir and 18 Noble Firs for a total of $1,342:
\[
d + 18n = 1342 \quad \text{(2)}
\]

Now we can solve this system of equations.

First, we can subtract equation (1) from equation (2) to eliminate \( d \):

\[
(d + 18n) - (d + 7n) = 1342 - 550
\]

This simplifies to:

\[
11n = 792
\]

Now, solve for \( n \):

\[
n = \frac{792}{11} = 72
\]

Now that we have the cost of a Noble Fir tree (\( n = 72 \)), we can substitute \( n \) back into equation (1) to find \( d \):

\[
d + 7(72) = 550
\]

This simplifies to:

\[
d + 504 = 550
\]

Subtract 504 from both sides:

\[
d = 550 - 504 = 46
\]

Now we have found the costs of each type of tree:

- A Douglas Fir costs \( 46 \) dollars.
- A Noble Fir costs \( 72 \) dollars.

Finally, fill in the blanks:

A Douglas Fir costs $ **46** and a Noble Fir costs $ **72**.