Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Percy and his mom are planting vegetables in their garden. Percy has finished planting $8$ rows of carrots so far and is planting new rows at a rate of $4$ rows per hour. His mom has finished $6$ rows of tomatoes and will continue planting at $5$ 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 many rows will they each have completed? How long will that take? $\newline$ Percy and his mom will each be done planting $\_\_\_$ rows in $\_\_\_$ hours.

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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

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

8

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

4

rows per hour. His mom has finished

6

rows of tomatoes and will continue planting at

5

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 many rows will they each have completed? How long will that take?

\newline

Percy and his mom will each be done planting

\_\_\_

rows in

\_\_\_

hours.

Define variables: Let's define the variables for the number of rows completed by Percy and his mom. Let $x$ be the number of hours they continue planting. Percy has already planted $8$ rows and plants at a rate of $4$ rows per hour. His mom has planted $6$ rows and plants at a rate of $5$ rows per hour. We want to find when they will have planted the same number of rows.
Write equations: Write the equations based on the given rates and initial conditions. For Percy, the total number of rows is $8 + 4x$ . For his mom, the total number of rows is $6 + 5x$ . We set these equal to each other to find when they will have the same number of rows. $\newline$ Percy's rows: $8 + 4x$ $\newline$ Mom's rows: $6 + 5x$ $\newline$ Equation: $8 + 4x = 6 + 5x$
Solve equation for x: Solve the equation for x. Subtract $4x$ from both sides to get the x terms on one side. $\newline$ $8 + 4x - 4x = 6 + 5x - 4x$ $\newline$ $8 = 6 + x$ $\newline$ Subtract $6$ from both sides to solve for x. $\newline$ $8 - 6 = x$ $\newline$ $x = 2$
Find rows completed: Now that we have the value of $x$ , which is the number of hours, we can find out how many rows each has completed. For Percy, plug $x = 2$ into his row equation: $\newline$ Percy's rows: $8 + 4(2) = 8 + 8 = 16$ $\newline$ For his mom, plug $x = 2$ into her row equation: $\newline$ Mom's rows: $6 + 5(2) = 6 + 10 = 16$
Check solution: Check the solution by plugging the values back into the original equations to ensure they are equal. $\newline$ Percy's rows: $8 + 4(2) = 16$ $\newline$ Mom's rows: $6 + 5(2) = 16$ $\newline$ Both are equal to $16$ , so the solution is correct.