Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Ben and his cousin Rachel are picking apples in their grandparents' orchard. Ben has filled $8$ baskets with apples and is filling them at a rate of $5$ baskets per hour. Rachel has $11$ full baskets and will continue picking at $2$ 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 long will that take? How much fruit will they have picked by then? $\newline$ In $\_\_$ hours, the cousins will each have filled $\_\_$ baskets with apples.

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Grade 8

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

Ben and his cousin Rachel are picking apples in their grandparents' orchard. Ben has filled

8

baskets with apples and is filling them at a rate of

5

baskets per hour. Rachel has

11

full baskets and will continue picking at

2

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 long will that take? How much fruit will they have picked by then?

\newline

\_\_

hours, the cousins will each have filled

\_\_

baskets with apples.

Set up equations: Let's set up the equations for Ben and Rachel. Ben starts with $8$ baskets and picks $5$ more per hour. Rachel starts with $11$ baskets and picks $2$ more per hour. We need to find when they have the same number of baskets. $\newline$ Ben's baskets: $B = 8 + 5t$ $\newline$ Rachel's baskets: $R = 11 + 2t$
Solve for t: Now, we solve for $t$ when $B = R$ . Setting the equations equal to each other: $\newline$ $8 + 5t = 11 + 2t$
Subtract and simplify: Subtract $2t$ from both sides to get: $\newline$ $8 + 3t = 11$
Divide by $3$ : Subtract $8$ from both sides: $\newline$ $3t = 3$
Plug in $t$ : Divide both sides by $3$ : $t = 1$
Check consistency: Plug $t = 1$ back into either original equation to find the number of baskets each has at that time. Using Ben's equation: $\newline$ $B = 8 + 5(1) = 13$ baskets
Check consistency: Plug $t = 1$ back into either original equation to find the number of baskets each has at that time. Using Ben's equation: $\newline$ $B = 8 + 5(1) = 13$ baskets Check Rachel's equation for consistency: $\newline$ $R = 11 + 2(1) = 13$ baskets