The solution to the system of equations is: x = 5. It will take 5 minutes for the two groups to clean the same number of rows.
Let's start by writing a system of equations. We know the following:
The Band Booster Club has already cleaned 5 rows of bleachers.
The Band Booster Club will continue to clean at a rate of 8 rows per minute.
The leadership class has completed 10 rows of bleachers.
The leadership class will continue working at 7 rows per minute.
We can use this information to write down the following equations:
# of rows cleaned by BBC = 5 + 8x
# of rows cleaned by LC = 10 + 7x
Once the two groups have cleaned the same number of rows, we can set these two equations equal to each other and solve for x.
5 + 8x = 10 + 7x
Subtracting 7x from both sides and subtracting 5 from both sides, we get:
x = 5
This means that it will take 5 minutes for the two groups to clean the same number of rows.
To graph these equations, we can use the following steps:
Convert the equations to y = mx + b form.
Plot the y-intercept for each line.
Find the slope of each line and use it to calculate additional points on the line.
Connect the points to form the lines.
The first equation, in y = mx + b form, is:
y = 8x + 5
The second equation, in y = mx + b form, is:
y = 7x + 10
The y-intercept for the Band Booster Club line is 5, so we plot a point at (0, 5).
The y-intercept for the leadership class line is 10, so we plot a point at (0, 10).
The slope of the Band Booster Club line is 8.
This means that for every 1 unit we move up on the y-axis, we also move 8 units to the right on the x-axis.
We can use this information to plot a second point on the line, such as (8, 61).
The slope of the leadership class line is 7. This means that for every 1 unit we move up on the y-axis, we also move 7 units to the right on the x-axis.
We can use this information to plot a second point on the line, such as (7, 67).
The x-coordinate of the point of intersection is 5, so it will take 5 minutes for the two groups to clean the same number of rows.
Therefore, the solution to the system of equations is: x = 5 .
Question
Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned 5 rows of bleachers and will continue to clean at a rate of 8 rows per minute. The leadership class has completed 10 rows and will continue working at 7 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? Write a system of equations, graph them, and type the solution.