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 $12$ Noble Fir trees, earning a total of $\$1,088$ . In the afternoon, they sold $9$ Douglas Fir trees and $12$ Noble Fir trees, earning a total of $\$864$ . How much does each type of tree cost? $\newline$ A Douglas Fir costs $\$\_\_\_\_$ and a Noble Fir costs $\$\_\_\_\_$ .

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Grade 8

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

12

Noble Fir trees, earning a total of

\$1,088

. In the afternoon, they sold

9

Douglas Fir trees and

12

Noble Fir trees, earning a total of

\$864

. How much does each type of tree cost?

\newline

A Douglas Fir costs

\$\_\_\_\_

and a Noble Fir costs

\$\_\_\_\_

Define variables: Let's define two variables: let $x$ be the cost of one Douglas Fir tree and $y$ be the cost of one Noble Fir tree. We can write two equations based on the information given: $\newline$ For the morning sales: $17x + 12y = 1088$ $\newline$ For the afternoon sales: $9x + 12y = 864$
Elimination method: To solve this system using elimination, we want to eliminate one of the variables. We can do this by subtracting the second equation from the first equation: $\newline$ $(17x + 12y) - (9x + 12y) = 1088 - 864$ $\newline$ This simplifies to: $\newline$ $17x - 9x + 12y - 12y = 1088 - 864$
Simplify equation: Now, we simplify the equation: $8x = 224$
Solve for x: Next, we solve for x by dividing both sides of the equation by $8$ : $\newline$ $x = \frac{224}{8}$ $\newline$ $x = 28$ $\newline$ So, a Douglas Fir tree costs $\$28$ .
Substitute $x$ back: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . We'll use the second equation: $\newline$ $9x + 12y = 864$ $\newline$ $9(28) + 12y = 864$ $\newline$ $252 + 12y = 864$
Solve for y: Subtract $252$ from both sides to solve for y: $\newline$ $12y = 864 - 252$ $\newline$ $12y = 612$
Solve for y: Subtract $252$ from both sides to solve for y: $\newline$ $12y = 864 - 252$ $\newline$ $12y = 612$ Finally, divide both sides by $12$ to find the value of $y$ : $\newline$ $y = 612 / 12$ $\newline$ $y = 51$ $\newline$ So, a Noble Fir tree costs $\$51$ .