Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ On the windowsill is a plant that is $19$ inches tall. It is growing $2$ inches per week. A second plant, which is $20$ inches tall, is on the coffee table. It is growing $1$ inch per week. Eventually the two plants will be the same height. How tall will the plants be? How many weeks will that take? $\newline$ Each plant will be $\_$ inches tall after $\_$ weeks.

Full solution

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

\newline

On the windowsill is a plant that is

19

inches tall. It is growing

2

inches per week. A second plant, which is

20

inches tall, is on the coffee table. It is growing

1

inch per week. Eventually the two plants will be the same height. How tall will the plants be? How many weeks will that take?

\newline

Each plant will be

\_

inches tall after

\_

weeks.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of weeks that have passed. $\newline$ Let $y$ be the height of the plants when they are the same height. $\newline$ We can write two equations to represent the growth of each plant: $\newline$ For the plant on the windowsill: $y = 19 + 2x$ (since it starts at $19$ inches and grows $2$ inches per week) $\newline$ For the plant on the coffee table: $y = 20 + x$ (since it starts at $20$ inches and grows $1$ inch per week) $\newline$ We want to find the value of $y$ and $x$ when the plants are the same height, so we set the two equations equal to each other: $\newline$ $19 + 2x = 20 + x$
Write Growth Equations: Now we solve for $x$ using the equation we just set up: $\newline$ $19 + 2x = 20 + x$ $\newline$ Subtract $x$ from both sides to get: $\newline$ $19 + x = 20$ $\newline$ Subtract $19$ from both sides to isolate $x$ : $\newline$ $x = 20 - 19$ $\newline$ $x = 1$ $\newline$ So, it will take $1$ week for the plants to be the same height.
Set Equations Equal: Next, we substitute $x = 1$ back into one of the original equations to find $y$ , the height of the plants when they are the same height. We can use either equation, but we'll use the first one: $\newline$ $y = 19 + 2x$ $\newline$ $y = 19 + 2(1)$ $\newline$ $y = 19 + 2$ $\newline$ $y = 21$ $\newline$ So, the plants will be $21$ inches tall after $1$ week.