Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Hanson and his mom are planting vegetables in their garden. Hanson has finished planting $9$ rows of carrots so far and is planting new rows at a rate of $2$ rows per hour. His mom has finished $7$ rows of tomatoes and will continue planting at $3$ rows per hour. Once the pair get to the point where they have finished the same number of rows, they will take a break and decide what to plant next. How long will that take? How many rows will they each have completed? $\newline$ In $\_$ hours, Hanson and his mom will each have planted $\_$ rows of vegetables.

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Solve a system of equations using any method: word problems

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

\newline

Hanson and his mom are planting vegetables in their garden. Hanson has finished planting

9

rows of carrots so far and is planting new rows at a rate of

2

rows per hour. His mom has finished

7

rows of tomatoes and will continue planting at

3

rows per hour. Once the pair get to the point where they have finished the same number of rows, they will take a break and decide what to plant next. How long will that take? How many rows will they each have completed?

\newline

\_

hours, Hanson and his mom will each have planted

\_

rows of vegetables.

Define Variables: Let's define the variables: Let $h$ be the number of hours it takes for Hanson and his mom to have planted the same number of rows. Let $r$ be the number of rows they each have completed.
Write First Equation: Write the first equation based on Hanson's progress. He has already planted $9$ rows and plants at a rate of $2$ rows per hour. So, the total number of rows he will have planted after $h$ hours is $9 + 2h$ .
First equation: $r = 9 + 2h$
Write Second Equation: Write the second equation based on his mom's progress. She has already planted $7$ rows and plants at a rate of $3$ rows per hour. So, the total number of rows she will have planted after $h$ hours is $7 + 3h$ . $\newline$ Second equation: $r = 7 + 3h$
Set Equations Equal: Since Hanson and his mom will take a break when they have finished the same number of rows, we can set the two equations equal to each other to find the value of $h$ . $9 + 2h = 7 + 3h$
Solve for h: Solve for h by subtracting $2h$ from both sides of the equation. $\newline$ $9 + 2h - 2h = 7 + 3h - 2h$ $\newline$ $9 = 7 + h$
Isolate $h$ : Now, subtract $7$ from both sides to isolate $h$ . $9 - 7 = 7 + h - 7$ $2 = h$
Substitute $h$ : We found that $h = 2$ . Now we need to find the number of rows $r$ they each have completed. We can substitute $h = 2$ into either the first or second equation. Let's use the first equation. $\newline$ $r = 9 + 2h$ $\newline$ $r = 9 + 2(2)$
Calculate r: Calculate the value of r. $\newline$ $r = 9 + 4$ $\newline$ $r = 13$
Final Result: We have found that in $2$ hours, Hanson and his mom will each have planted $13$ rows of vegetables.