Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Two groups of workers are painting a bridge in the bay. The first group is responsible for painting the north side of the bridge, and the second group is responsible for painting the south side of the bridge. The first group has already painted $5$ kilometers of the bridge and is painting $2$ additional kilometers per day. The second group has already painted $2$ kilometers of the bridge and is painting $3$ additional kilometers per day. After a while, the two groups will have painted the same amount of the bridge. How long will that take? How much of the bridge will each group have painted? $\newline$ In $\_$ days, both groups of workers will have painted $\_$ kilometers of the bridge.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Two groups of workers are painting a bridge in the bay. The first group is responsible for painting the north side of the bridge, and the second group is responsible for painting the south side of the bridge. The first group has already painted

5

kilometers of the bridge and is painting

2

additional kilometers per day. The second group has already painted

2

kilometers of the bridge and is painting

3

additional kilometers per day. After a while, the two groups will have painted the same amount of the bridge. How long will that take? How much of the bridge will each group have painted?

\newline

\_

days, both groups of workers will have painted

\_

kilometers of the bridge.

Define Variables: Let's denote the number of days after which both groups will have painted the same amount of the bridge as $d$ days. The first group has already painted $5$ kilometers and paints $2$ kilometers per day. The second group has already painted $2$ kilometers and paints $3$ kilometers per day.
Write Equations: We can write two equations to represent the total distance painted by each group after $d$ days. For the first group, the total distance painted will be $5$ kilometers plus $2$ kilometers per day times the number of days. For the second group, it will be $2$ kilometers plus $3$ kilometers per day times the number of days. $\newline$ First group: $5 + 2d$ $\newline$ Second group: $2 + 3d$
Set Equations Equal: Since we are looking for the point where both groups have painted the same amount, we can set the two expressions equal to each other to find $d$ . $\newline$ $5 + 2d = 2 + 3d$
Solve for d: Now, we solve for ＂d＂. Subtract $2d$ from both sides to get: $\newline$ $5 = 2 + d$
Find Total Distance: Subtract $2$ from both sides to isolate ＂ $d$ ＂: $\newline$ $5 - 2 = d$ $\newline$ $3 = d$
Find Total Distance: Subtract $2$ from both sides to isolate ＂ $d$ ＂: $\newline$ $5 - 2 = d$ $\newline$ $3 = d$ Now that we have the number of days, we can find out how much each group has painted. We'll plug ＂ $d$ ＂ back into one of the original equations. $\newline$ For the first group: $5 + 2d = 5 + 2(3) = 5 + 6 = 11$ kilometers $\newline$ For the second group: $2 + 3d = 2 + 3(3) = 2 + 9 = 11$ kilometers