Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Erin has been planting young trees in her garden. The maple tree that is $28$ inches tall is growing $2$ inches per month, whereas the oak tree that is $20$ inches tall is growing $3$ inches per month. In a few months, the two trees will be the same height. How long will that take? What will that height be? $\newline$ In $\_$ months, the two trees will both be $\_$ inches tall.

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Solve a system of equations using substitution: word problems

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

\newline

Erin has been planting young trees in her garden. The maple tree that is

28

inches tall is growing

2

inches per month, whereas the oak tree that is

20

inches tall is growing

3

inches per month. In a few months, the two trees will be the same height. How long will that take? What will that height be?

\newline

\_

months, the two trees will both be

\_

inches tall.

Define variables: Define variables for the heights of the trees and the time in months. $\newline$ Let's use $M$ to represent the height of the maple tree and $O$ to represent the height of the oak tree. Let $t$ be the time in months after which the trees will be the same height.
Write equations: Write the equations based on the given information. $\newline$ The maple tree is initially $28$ inches tall and grows $2$ inches per month. So, its height after $t$ months will be $M = 28 + 2t$ . $\newline$ The oak tree is initially $20$ inches tall and grows $3$ inches per month. So, its height after $t$ months will be $O = 20 + 3t$ . $\newline$ We want to find the time $t$ when $M = O$ .
Set up system: Set up the system of equations. $\newline$ We have two equations now: $\newline$ $1$ ) $M = 28 + 2t$ $\newline$ $2$ ) $O = 20 + 3t$ $\newline$ And since $M = O$ at the time we are looking for, we can set the right sides of these equations equal to each other: $\newline$ $28 + 2t = 20 + 3t$
Solve using substitution: Solve the system using substitution. $\newline$ Since we have $M = O$ , we can substitute the expression for $M$ from the first equation into the second equation: $\newline$ $28 + 2t = 20 + 3t$ $\newline$ Now, we solve for $t$ .
Rearrange to isolate: Rearrange the equation to isolate $t$ . $\newline$ Subtract $2t$ from both sides: $\newline$ $28 + 2t - 2t = 20 + 3t - 2t$ $\newline$ $28 = 20 + t$ $\newline$ Now, subtract $20$ from both sides: $\newline$ $28 - 20 = t$ $\newline$ $8 = t$
Find height: Find the height of the trees when they are the same. $\newline$ We know $t = 8$ months. Now we need to substitute $t$ back into either the first or second equation to find the height. Let's use the first equation: $\newline$ $M = 28 + 2t$ $\newline$ $M = 28 + 2(8)$ $\newline$ $M = 28 + 16$ $\newline$ $M = 44$