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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 $
.

1 answer

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.