Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Steven and his cousin Carly are picking apples in their grandparents' orchard. Steven has filled $15$ baskets with apples and is filling them at a rate of $4$ baskets per hour. Carly has $11$ full baskets and will continue picking at $8$ 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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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

Steven and his cousin Carly are picking apples in their grandparents' orchard. Steven has filled

15

baskets with apples and is filling them at a rate of

4

baskets per hour. Carly has

11

full baskets and will continue picking at

8

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.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of hours it takes for Steven and Carly to fill the same number of baskets. $\newline$ Let $y$ be the total number of baskets each has filled at that time. $\newline$ Steven's equation: $\newline$ Steven starts with $15$ baskets and fills them at a rate of $4$ baskets per hour. $\newline$ $y = 4x + 15$
Steven's Equation: Carly's equation: $\newline$ Carly starts with $11$ baskets and fills them at a rate of $8$ baskets per hour. $\newline$ $y = 8x + 11$
Carly's Equation: Now we have a system of equations: $\newline$ $1$ ) $y = 4x + 15$ $\newline$ $2$ ) $y = 8x + 11$ $\newline$ We can use substitution to solve for $x$ by setting the two equations equal to each other since they both equal $y$ . $\newline$ $4x + 15 = 8x + 11$
System of Equations: Solve for $x$ :
Subtract $4x$ from both sides:
$4x + 15 - 4x = 8x + 11 - 4x$
$15 = 4x + 11$

Subtract $11$ from both sides:
$15 - 11 = 4x + 11 - 11$
$4 = 4x$

Divide both sides by $4$ :
$\frac{4}{4} = \frac{4x}{4}$
$1 = x$
Solve for x: Now that we have the value of $x$ , we can substitute it back into either equation to find $y$ . We'll use Steven's equation: $\newline$ $y = 4x + 15$ $\newline$ $y = 4(1) + 15$ $\newline$ $y = 4 + 15$ $\newline$ $y = 19$