Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A boy scout troop is selling Christmas trees at a local tree lot. In the morning, they sold $17$ Douglas Fir trees and $23$ Noble Fir trees, earning a total of $\$2,009$ . In the afternoon, they sold $8$ Douglas Fir trees and $5$ Noble Fir trees, earning a total of $\$596$ . How much does each type of tree cost? $\newline$ A Douglas Fir costs _ and a Noble Fir costs $\$$ _.

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Algebra 1

Solve a system of equations using elimination: word problems

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

\newline

A boy scout troop is selling Christmas trees at a local tree lot. In the morning, they sold

17

Douglas Fir trees and

23

Noble Fir trees, earning a total of

\$2,009

. In the afternoon, they sold

8

Douglas Fir trees and

5

Noble Fir trees, earning a total of

\$596

. How much does each type of tree cost?

\newline

A Douglas Fir costs _____ and a Noble Fir costs

\$

_____.

Identify Equations: Identify the equations based on the given information. $\newline$ Morning sales: $17$ Douglas Fir trees $(D)$ and $23$ Noble Fir trees $(N)$ for $\$2009$ . $\newline$ Afternoon sales: $8$ Douglas Fir trees $(D)$ and $5$ Noble Fir trees $(N)$ for $\$596$ . $\newline$ Translate this information into two equations: $\newline$ $(D)$ $0$ $\newline$ $(D)$ $1$
Multiply Second Equation: Multiply the second equation by a number that will allow us to eliminate one of the variables when we subtract the equations from each other. We can multiply the second equation by $-2.1$ (which is $-\frac{21}{10}$ ) to eliminate the variable $D$ , as $17$ is close to $2.1$ times $8$ . $\newline$ $-2.1(8D + 5N) = -2.1(596)$
Perform Multiplication: Perform the multiplication from Step $2$ . $\newline$ $-16.8D - 10.5N = -1251.6$ $\newline$ Now we have a new equation that we can use to eliminate $D$ .
Add Equations: Add the new equation from Step $3$ to the first equation to eliminate $D$ . $\newline$ $17D + 23N = 2009$ $\newline$ $-16.8D - 10.5N = -1251.6$ $\newline$ Adding these equations, we get: $\newline$ $0.2D + 12.5N = 757.4$
Solve for N: Solve for N by dividing both sides of the equation by $12.5$ . $\newline$ $\frac{0.2D}{12.5} + \frac{12.5N}{12.5} = \frac{757.4}{12.5}$ $\newline$ $0.016D + N = 60.592$ $\newline$ This equation does not look correct. We should have a whole number for the price of a tree. There seems to be a mistake in the multiplication or the addition of the equations. Let's go back and check our calculations.