Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Darnel has been planting young trees in her garden. The maple tree that is $57$ centimeters tall is growing $7$ centimeters per month, whereas the oak tree that is $63$ centimeters tall is growing $5$ centimeters 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 $\_$ centimeters 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

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

57

centimeters tall is growing

7

centimeters per month, whereas the oak tree that is

63

centimeters tall is growing

5

centimeters 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

\_

centimeters tall in

\_

months.

Introduction: Let $x$ represent the number of months, and $y$ represent the height of the trees in centimeters after $x$ months. $\newline$ For the maple tree: $\newline$ Initial height: $57$ centimeters $\newline$ Growth rate per month: $7$ centimeters/month $\newline$ Equation based on the given information: $\newline$ Height after $x$ months = initial height + (growth rate * number of months) $\newline$ $y = 7x + 57$ $\newline$ For the maple tree, the equation is: $y = 7x + 57$
Maple Tree Information: For the oak tree: $\newline$ Initial height: $63$ centimeters $\newline$ Growth rate per month: $5$ centimeters/month $\newline$ Equation based on the given information: $\newline$ Height after $x$ months = initial height + (growth rate * number of months) $\newline$ $y = 5x + 63$ $\newline$ For the oak tree, the equation is: $y = 5x + 63$
Oak Tree Information: System of equations: $\newline$ $y = 7x + 57$ $\newline$ $y = 5x + 63$ $\newline$ To find the point where the trees are the same height, set the two equations equal to each other: $\newline$ $7x + 57 = 5x + 63$
System of Equations: Solve for $x$ : $\newline$ Subtract $5x$ from both sides of the equation: $\newline$ $7x + 57 - 5x = 5x + 63 - 5x$ $\newline$ $2x + 57 = 63$ $\newline$ Now, subtract $57$ from both sides: $\newline$ $2x + 57 - 57 = 63 - 57$ $\newline$ $2x = 6$ $\newline$ Divide both sides by $2$ to isolate $x$ : $\newline$ $\frac{2x}{2} = \frac{6}{2}$ $\newline$ $5x$ $0$
Solving for x: Find the value of $y$ by substituting $x$ into one of the system of equations: $\newline$ Substitute $3$ for $x$ in $y = 7x + 57$ : $\newline$ $y = 7(3) + 57 = 21 + 57 = 78$ $\newline$ So, $y = 78$