Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Bobby has been planting young trees in his garden. The maple tree that is $14$ inches tall is growing $2$ inches per month, whereas the oak tree that is $10$ inches tall is growing $3$ inches per month. In a few months, the two trees will be the same height. What will that height be? How long will that take? $\newline$ The two trees will both be $\_$ inches tall in $\_$ months.

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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

Bobby has been planting young trees in his garden. The maple tree that is

14

inches tall is growing

2

inches per month, whereas the oak tree that is

10

inches tall is growing

3

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

\newline

The two trees will both be

\_

inches tall in

\_

months.

Maple Tree Equation: Let $x$ represent the number of months and $y$ represent the height of the trees in inches. $\newline$ For the maple tree: $\newline$ Initial height: $14$ inches $\newline$ Growth rate: $2$ inches per month $\newline$ Equation based on the given information: $\newline$ Height in inches = growth rate per month $*$ number of months + initial height $\newline$ $y = 2x + 14$ $\newline$ For the maple tree, the equation is: $y = 2x + 14$
Oak Tree Equation: For the oak tree: $\newline$ Initial height: $10$ inches $\newline$ Growth rate: $3$ inches per month $\newline$ Equation based on the given information: $\newline$ Height in inches = growth rate per month $\times$ number of months + initial height $\newline$ $y = 3x + 10$ $\newline$ For the oak tree, the equation is: $y = 3x + 10$
System of Equations: System of equations: $\newline$ $y = 2x + 14$ $\newline$ $y = 3x + 10$ $\newline$ To find when the trees will be the same height, set the two equations equal to each other. $\newline$ $2x + 14 = 3x + 10$
Solving for x: Solve for x by isolating the variable. $\newline$ Subtract $2x$ from both sides of the equation: $\newline$ $2x + 14 - 2x = 3x + 10 - 2x$ $\newline$ $14 = x + 10$ $\newline$ Now, subtract $10$ from both sides: $\newline$ $14 - 10 = x + 10 - 10$ $\newline$ $4 = x$ $\newline$ So, $x = 4$
Substitute $x$ into Equation: Now that we have $x = 4$ , we can find the height $y$ by substituting $x$ into one of the original equations. $\newline$ Substitute $4$ for $x$ in $y = 2x + 14$ : $\newline$ $y = 2(4) + 14$ $\newline$ $y = 8 + 14$ $\newline$ $y = 22$ $\newline$ So, $y = 22$
Final Result: We have found that $x = 4$ and $y = 22$ . This means that in $4$ months, both the maple and oak trees will be $22$ inches tall.