# How do I solve the following problem using a system of equations and the elimination method?A plumber and a builder were employed to fit a new bath, each working a different number of hours. The...

How do I solve the following problem using a system of equations and the elimination method?

A plumber and a builder were employed to fit a new bath, each working a different number of hours. The plumber earns $35 per hour, and the builder earns $28 per hour. Together they were paid $330.75, but the plumber earned $106.75 more than the builder. How many hours did each work?

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### 2 Answers

Let the number of hours the plumber worked be P and the number of hours the builder worked be B.

The total amount earned is by the plumber at the rate of $35 per hour is 35P. The total amount earned by the builder at the rate of $28 per hour is 25P.

Together they were paid $330.75. This gives:

35P + 28B = 330.75 ...(1)

The plumber earned $106.75 more than the builder. This gives:

35P = 28B + 106.75 ...(2)

The system of equations (1) and (2) has to be solved by elimination.

First, let's eliminate P

(1) - (2)

=> 35P + 28B - 35P = 330.75 - 28B - 106.75

=> 28B + 28B = 224

=> B = 224/56 = 4

Now, to eliminate B

(1) + (2)

=> 35P + 28B + 35P = 330.75 + 28B + 106.75

=> 70P + 28B - 28B = 437.5

=> P = 437.5/70

=> P = 6.25

**The plumber worked for 6.25 hours and the builder worked for 4 hours.**

Start by choosing variables to represent the number of hours the plumber and the builder each worked:

Let x = the number of hours the plumber worked.

Let y = the number of hours the builder worked.

Now use the information in the problem to write two equations involving x and y.

$35x represents the amount of money the plumber earned, and $28y represents the amount of money the builder earned. Since together they earned $330.75, an equation representing this relationship is

35x + 28y = 330.75

An equation representing the fact that the plumber earned $106.75 more than the builder is

35x - 28y = 106.75

These two equations make up the system of equations you need to solve. Using elimination, the steps are as follows.

35x + 28y = 330.75 Subtract the bottom equation from the

35x - 28y = 106.75 top equation to eliminate the x terms.

_________________

56y = 224.00

Divide both sides by 56 to get y = 4.

This is the number of hours the builder worked.

Substitute 4 for y in either of the two equations and solve for x. Choosing the first equation gives

35x + 28(4) = 330.75 Now perform the multiplication to get

35x + 112 = 330.75 Subtracting 112 from both sides gives

35x = 218.75 Now divide both sides by 35

x = 6.25, or 6 1/4 This is how many hours the plumber worked.

**The plumber worked 6 1/4 hours and the builder worked 4 hours.**