Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned $12$ rows of bleachers and will continue to clean at a rate of $7$ rows per minute. The leadership class has completed $6$ rows and will continue working at $8$ rows per minute. Once the two groups get to the point where they have cleaned the same number of rows, they will take a break and decide how to split up the remaining work. How long will that take? How many minutes will each group have cleaned by then? $\newline$ In $\_$ minutes, the groups will each cleaned $\_$ rows each.

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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

Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned

12

rows of bleachers and will continue to clean at a rate of

7

rows per minute. The leadership class has completed

6

rows and will continue working at

8

rows per minute. Once the two groups get to the point where they have cleaned the same number of rows, they will take a break and decide how to split up the remaining work. How long will that take? How many minutes will each group have cleaned by then?

\newline

\_

minutes, the groups will each cleaned

\_

rows each.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of minutes both groups will work until they have cleaned the same number of rows. $\newline$ Let $y$ be the total number of rows cleaned by each group when they take a break. $\newline$ Band Booster Club's rate of cleaning is $7$ rows per minute, and they have already cleaned $12$ rows. $\newline$ The equation for the Band Booster Club is: $\newline$ $y = 7x + 12$
Band Booster Club: The leadership class's rate of cleaning is $8$ rows per minute, and they have already cleaned $6$ rows. $\newline$ The equation for the leadership class is: $\newline$ $y = 8x + 6$
Leadership Class: Now we have two equations: $\newline$ $1$ ) $y = 7x + 12$ (Band Booster Club) $\newline$ $2$ ) $y = 8x + 6$ (Leadership class) $\newline$ We can use substitution to find the value of $x$ by setting the two equations equal to each other since $y$ represents the same total number of rows cleaned by each group at the time they take a break. $\newline$ $7x + 12 = 8x + 6$
Substitution: Solving for $x$ : $7x + 12 = 8x + 6$ Subtract $7x$ from both sides: $12 = x + 6$ Subtract $6$ from both sides: $x = 12 - 6$ $x = 6$
Solving for x: Now that we have the value of $x$ , we can substitute it back into either of the original equations to find $y$ . Let's use the first equation: $\newline$ $y = 7x + 12$ $\newline$ $y = 7(6) + 12$ $\newline$ $y = 42 + 12$ $\newline$ $y = 54$