Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Finn works in an amusement park and is helping decorate it with strands of lights. This morning, he used a total of $60$ strands of lights to decorate $5$ bushes and $3$ trees. This afternoon, he strung lights on $5$ bushes and $2$ trees, using a total of $50$ strands. Assuming that all bushes are decorated one way and all trees are decorated another, how many strands did Finn use on each? $\newline$ Finn decorated every bush with $\_\_\_$ strands of lights and every tree with $\_\_\_$ strands.

Full solution

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

\newline

Finn works in an amusement park and is helping decorate it with strands of lights. This morning, he used a total of

60

strands of lights to decorate

5

bushes and

3

trees. This afternoon, he strung lights on

5

bushes and

2

trees, using a total of

50

strands. Assuming that all bushes are decorated one way and all trees are decorated another, how many strands did Finn use on each?

\newline

Finn decorated every bush with

\_\_\_

strands of lights and every tree with

\_\_\_

strands.

Define variables: Let's define variables for the number of strands used per bush and per tree. $\newline$ Let $b =$ number of strands per bush $\newline$ Let $t =$ number of strands per tree
Write first equation: Write the first equation based on the morning decoration. $\newline$ $5$ bushes and $3$ trees use $60$ strands. $\newline$ $5b + 3t = 60$
Write second equation: Write the second equation based on the afternoon decoration. $\newline$ $5$ bushes and $2$ trees use $50$ strands. $\newline$ $5b + 2t = 50$
Elimination method: Solve the system of equations using the substitution or elimination method. We will use the elimination method. $\newline$ To eliminate one of the variables, we can multiply the second equation by $-1.5$ and add it to the first equation. $\newline$ $-1.5(5b + 2t) = -1.5(50)$ $\newline$ $-7.5b - 3t = -75$ $\newline$ Now add this to the first equation: $\newline$ $5b + 3t = 60$ $\newline$ $-7.5b - 3t = -75$ $\newline$ ----------------- $\newline$ $-2.5b = -15$
Solve for b: Solve for b. $\newline$ $-2.5b = -15$ $\newline$ Divide both sides by $-2.5$ to find b. $\newline$ $b = \frac{-15}{-2.5}$ $\newline$ $b = 6$
Substitute to solve for t: Substitute the value of $b$ back into one of the original equations to solve for $t$ . Using the second equation: $5b + 2t = 50$ $5(6) + 2t = 50$ $30 + 2t = 50$ $2t = 50 - 30$ $2t = 20$ Divide both sides by $2$ to find $t$ . $t = \frac{20}{2}$ $t$ $0$