Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mark and his cousin Leah are picking apples in their grandparents' orchard. Mark has filled $6$ baskets with apples and is filling them at a rate of $4$ baskets per hour. Leah has $12$ full baskets and will continue picking at $3$ baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How much fruit will they have picked by then? How long will that take? $\newline$ Mark and Leah will each have filled $\_\_$ baskets in $\_\_$ hours.

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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

Mark and his cousin Leah are picking apples in their grandparents' orchard. Mark has filled

6

baskets with apples and is filling them at a rate of

4

baskets per hour. Leah has

12

full baskets and will continue picking at

3

baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How much fruit will they have picked by then? How long will that take?

\newline

Mark and Leah will each have filled

\_\_

baskets in

\_\_

hours.

Define Variables: Let's define the variables for the number of baskets Mark and Leah will have filled after a certain number of hours. Let $x$ represent the number of hours that have passed since they started picking apples. Mark starts with $6$ baskets and fills $4$ baskets per hour, so the total number of baskets he will have filled after $x$ hours is $6 + 4x$ . Leah starts with $12$ baskets and fills $3$ baskets per hour, so the total number of baskets she will have filled after $x$ hours is $12 + 3x$ . We want to find the point where they have filled the same number of baskets, so we set the two expressions equal to each other to create the first equation of our system: $6 + 4x = 12 + 3x$
Set Up Equation: Now we need to solve for $x$ . To do this, we will isolate $x$ on one side of the equation. We can start by subtracting $3x$ from both sides to get rid of the $x$ on the right side of the equation: $\newline$ $6 + 4x - 3x = 12 + 3x - 3x$ $\newline$ This simplifies to: $\newline$ $6 + x = 12$
Solve for x: Next, we subtract $6$ from both sides to solve for x: $\newline$ $6 + x - 6 = 12 - 6$ $\newline$ This simplifies to: $\newline$ $x = 6$ $\newline$ So, after $6$ hours, they will have filled the same number of baskets.
Calculate Mark's Baskets: Now we need to determine how many baskets each person will have filled after $6$ hours. For Mark: $\newline$ $6 + 4x = 6 + 4(6) = 6 + 24 = 30$ $\newline$ Mark will have filled $30$ baskets.
Calculate Leah's Baskets: For Leah: $12 + 3x = 12 + 3(6) = 12 + 18 = 30$ Leah will also have filled $30$ baskets.
Final Answer: We have found that both Mark and Leah will have filled $30$ baskets each after $6$ hours. This answers the question prompt.