Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Isaac works in an amusement park and is helping decorate it with strands of lights. This morning, he used a total of $43$ strands of lights to decorate $5$ bushes and $2$ trees. This afternoon, he strung lights on $5$ bushes and $3$ trees, using a total of $52$ strands. Assuming that all bushes are decorated one way and all trees are decorated another, how many strands did Isaac use on each? $\newline$ Isaac decorated every bush with $\_$ strands of lights and every tree with $\_$ strands.

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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

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

43

strands of lights to decorate

5

bushes and

2

trees. This afternoon, he strung lights on

5

bushes and

3

trees, using a total of

52

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

\newline

Isaac decorated every bush with

\_

strands of lights and every tree with

\_

strands.

Define Variables: Let's define two variables: let $b$ represent the number of strands used for each bush, and $t$ represent the number of strands used for each tree. We can then write two equations based on the information given: $\newline$ $1$ . For the morning: $5b + 2t = 43$ ( $5$ bushes and $2$ trees used a total of $43$ strands) $\newline$ $2$ . For the afternoon: $5b + 3t = 52$ ( $5$ bushes and $3$ trees used a total of $52$ strands) $\newline$ These two equations form our system of equations.
Write Equations: To solve this system using elimination, we want to eliminate one of the variables. We can do this by multiplying the first equation by $-1$ and then adding it to the second equation to eliminate $b$ . $\newline$ $-1\times(5b + 2t) = -1\times43$ $\newline$ $-5b - 2t = -43$ $\newline$ Now we add this to the second equation: $\newline$ $(5b + 3t) + (-5b - 2t) = 52 + (-43)$
Eliminate Variable: After adding the equations, the $b$ terms cancel out, and we are left with: $\newline$ $3t - 2t = 52 - 43$ $\newline$ $t = 9$ $\newline$ So, Isaac used $9$ strands of lights for each tree.
Solve for t: Now that we know the value of $t$ , we can substitute it back into one of the original equations to solve for $b$ . We'll use the first equation: $\newline$ $5b + 2t = 43$ $\newline$ $5b + 2(9) = 43$ $\newline$ $5b + 18 = 43$
Substitute t: Subtract $18$ from both sides to solve for $b$ : $\newline$ $5b = 43 - 18$ $\newline$ $5b = 25$ $\newline$ Now, divide both sides by $5$ : $\newline$ $b = 25 / 5$ $\newline$ $b = 5$ $\newline$ So, Isaac used $5$ strands of lights for each bush.