Question

write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
wesley is a plumber, and his coworker gordon is an electrician. wesley charges customers a fee of $151 just to come to their houses and then $1 per minute that he is there. gordon also charges a fee of $45 for a home visit, plus an additional $3 per minute. last week the coworkers went to a job site together, spent the same amount of time working, and earned the same amount. how much did each one earn? how much time did each one spend working?
wesley and gordon each earned $ by working for minutes.

Explanation:

Step1: Set up equations

Let \(x\) be the number of minutes each spent working.
For Wesley: The total amount he earned \(y = 151+3x\) (a fixed fee of \(151\) plus \(3\) per minute).
For Gordon: The total amount he earned \(y = 45 + 1x\) (a fixed fee of \(45\) plus \(1\) per minute).

Step2: Solve the system of equations

Since they earned the same amount, we set the two equations equal to each other:
\(151 + 3x=45+x\)
Subtract \(x\) from both sides:
\(151+3x - x=45+x - x\)
\(151 + 2x=45\)
Subtract \(151\) from both sides:
\(2x=45 - 151\)
\(2x=- 106\)
Divide both sides by \(2\):
\(x=\frac{-106}{2}=- 53\) (This is wrong, we made a mistake in the setup. Wait, no, actually, we should have set up the equations as Wesley's total \(y = 151+1\times x\) (wait no, re - read the problem).
Wait, Wesley: fee of \(151\) to come, plus \(1\) per minute. Gordon: fee of \(45\), plus \(3\) per minute.
So the correct equations are \(y = 151+x\) (Wesley) and \(y = 45 + 3x\) (Gordon)
Set \(151+x=45+3x\)
Subtract \(x\) from both sides: \(151=45 + 3x-x\)
\(151=45+2x\)
Subtract \(45\) from both sides: \(2x=151 - 45\)
\(2x = 106\)
Divide by \(2\): \(x = 53\)
Substitute \(x = 53\) into Wesley's equation \(y=151 + 53=204\)

Answer:

Wesley and Gordon each earned \(\$204\) by working for \(53\) minutes.