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?
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.