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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned rows of bleachers and will continue to clean at a rate of rows per minute. The leadership class has completed rows and will continue working at 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?In minutes, the groups will each cleaned rows each.
Full solution
Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned rows of bleachers and will continue to clean at a rate of rows per minute. The leadership class has completed rows and will continue working at 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?In minutes, the groups will each cleaned rows each.
- Define Variables: Let's define the variables:Let be the number of minutes both groups will work until they have cleaned the same number of rows.Let be the total number of rows cleaned by each group when they take a break.Band Booster Club's equation:They have already cleaned rows and will clean at a rate of rows per minute.So, their total rows cleaned after minutes will be .Leadership class's equation:They have completed rows and will continue working at rows per minute.So, their total rows cleaned after minutes will be .We need to find the values of and when both equations are equal, as that's when both groups will have cleaned the same number of rows.
- Equations Setup: Now we set up the system of equations:Band Booster Club: Leadership class: To solve for , we can use substitution by setting the two equations equal to each other since they both equal at the point of intersection.
- Solve for x: Now we solve for x:Subtract from both sides:Subtract from both sides:So, it will take minutes for both groups to have cleaned the same number of rows.
- Find y: Now we need to find , the number of rows cleaned by each group after minutes.We can substitute into either of the original equations. Let's use the Band Booster Club's equation:So, after minutes, each group will have cleaned rows each.
