Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ The volleyball team and the wrestling team at Arcadia High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $\$3$ per car. In addition, they have already brought in $\$43$ from past fundraisers. The wrestling team has raised $\$93$ in the past, and they are making $\$1$ per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take? $\newline$ Each team will have raised a total of $\$\_\_\_\_$ after washing $\_\_\_\_$ cars.

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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

The volleyball team and the wrestling team at Arcadia High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets

\$3

per car. In addition, they have already brought in

\$43

from past fundraisers. The wrestling team has raised

\$93

in the past, and they are making

\$1

per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?

\newline

Each team will have raised a total of

\$\_\_\_\_

after washing

\_\_\_\_

cars.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of cars washed. $\newline$ Let $V$ be the total amount raised by the volleyball team. $\newline$ Let $W$ be the total amount raised by the wrestling team. $\newline$ We are given that the volleyball team gets $\$3$ per car plus an additional $\$43$ from past fundraisers. So, the total amount raised by the volleyball team can be expressed as: $\newline$ $V = 3x + 43$
Calculate Total Amount Raised: For the wrestling team, we are given that they make $\$1$ per car plus $\$93$ from past fundraisers. So, the total amount raised by the wrestling team can be expressed as: $\newline$ $W = x + 93$
Set Equations Equal: We are told that after washing a certain number of cars, each team will have raised the same amount in total. This means that $V = W$ . So we can set the two expressions equal to each other to find the number of cars washed: $3x + 43 = x + 93$
Solve for x: Now we will solve for x using substitution. First, we'll subtract $x$ from both sides to get the x terms on one side: $\newline$ $3x + 43 - x = x + 93 - x$ $\newline$ $2x + 43 = 93$
Substitute x Value: Next, we'll subtract $43$ from both sides to solve for $x$ : $\newline$ $2x + 43 - 43 = 93 - 43$ $\newline$ $2x = 50$
Find Total Amount Raised: Finally, we'll divide both sides by $2$ to find the value of $x$ : $\frac{2x}{2} = \frac{50}{2}$ $x = 25$
Final Result: Now that we have the number of cars washed $x = 25$ , we can find the total amount raised by each team. Since the teams will have raised the same amount, we can use either the volleyball team's or the wrestling team's equation to find the total amount. We'll use the volleyball team's equation: $\newline$ $V = 3x + 43$ $\newline$ $V = 3(25) + 43$ $\newline$ $V = 75 + 43$ $\newline$ $V = 118$
Final Result: Now that we have the number of cars washed $x = 25$ , we can find the total amount raised by each team. Since the teams will have raised the same amount, we can use either the volleyball team's or the wrestling team's equation to find the total amount. We'll use the volleyball team's equation: $\newline$ $V = 3x + 43$ $\newline$ $V = 3(25) + 43$ $\newline$ $V = 75 + 43$ $\newline$ $V = 118$ We have found that each team will have raised a total of $\$118$ after washing $25$ cars.