A plumber, electrician and painter were all hired to work on 3 different house. For Adam's house, a plumber worked a total of 35 hours, an electrician worked 42 hours and a painter needed 29 hrs to complete the job. the total cost of the three workers for adam's house was $2,570.
A second house they worked on was the watson home. in this house each of the three workers needed 28 hours to complete their part of the job. The total cost of the work on the watson home was $1,960.
A third house belonged to the Miller family. The electrician worked 24 hours, the plumber 30 hours and the painter worked 25 hours. the total for this home was $1,845.
** Using the information from these three homes, how much did each of the workers charge per hour?
Please help I don't know how to solve this question
When I solved the system of equations I got M=25, E=30 and N=15 and when I substituted into each equation it worked, thank you so much!
First set up equation for each house to put together a system of 3 linear equations.
You can then solve by the method you have learned.
Finally post your answer for checking as necessary.
I will start the setting up of the equations.
Let
M=pluMber's hours
E=electrician's hours
N=paiNter's hours
For Adam's house:
35M+42E+29N=2570 ...(1) Adam's house
28M+28E+28N=1960 but can be simplified to
M+E+N=70 ...(2) Watson's house
30M+24E+25N=1845 ...(3) Miller's home
Note that the order is different in Miller's home, but rearranged to the same initial order.
The system of equations to be solved will then be:
35M+42E+29N=2570 ...(1)
M+E+N=70 ...........(2)
30M+24E+25N=1845 ...(3)
You can solve the system of equations using the method(s) you have learned, such as Gauss elimination, Cramer's rule, iterations, etc.
Post your answer for a check if you wish, or you can substitute the solution into each equation to check.
To find out how much each worker charged per hour, we can apply the following steps:
Step 1: Assign variables to unknowns
Let's assign the cost per hour for the plumber as P, for the electrician as E, and for the painter as R.
Step 2: Set up equations based on the given information
We can set up three equations based on the information given:
Equation 1: 35P + 42E + 29R = 2570 (Adam's house)
Equation 2: 28P + 28E + 28R = 1960 (Watson's house)
Equation 3: 30P + 24E + 25R = 1845 (Miller's house)
Step 3: Solve the system of equations
To solve this system of equations, we can use matrix algebra or substitution. We'll use the substitution method.
From Equation 2, we can isolate P:
28P = 1960 - 28E - 28R
P = (1960 - 28E - 28R)/28
P = 70 - E - R
Now substitute P in Equations 1 and 3:
Equation 1: 35(70 - E - R) + 42E + 29R = 2570
Equation 3: 30(70 - E - R) + 24E + 25R = 1845
Simplify the equations:
2450 - 35E - 35R + 42E + 29R = 2570
2100 - 30E - 30R + 24E + 25R = 1845
Combine like terms:
-6E - 6R = 120
-6E - 5R = -255
Multiply the second equation by -6:
36E + 30R = 1530
Combine the new equation with the first equation:
36E + 30R = 1530
-6E - 6R = 120
Subtract the second equation from the first equation:
42R = 1410
R = 1410/42
R = 33.57
Now substitute R = 33.57 back into the second equation:
-6E - 5(33.57) = -255
Simplify and solve for E:
-6E - 167.85 = -255
-6E = -87.15
E = (-87.15)/-6
E = 14.525
Finally, substitute R = 33.57 and E = 14.525 into the first equation:
35P + 42(14.525) + 29(33.57) = 2570
Simplify and solve for P:
35P + 615.15 + 974.13 = 2570
35P = 2570 - 615.15 - 974.13
35P = 980.72
P = 980.72/35
P ≈ 27.89
Therefore, the plumber charged approximately $27.89 per hour, the electrician charged approximately $14.53 per hour, and the painter charged approximately $33.57 per hour.
To find out how much each worker charged per hour, we need to divide the total cost of the work on each house by the number of hours worked by each worker. Let's calculate it step by step:
For Adam's house:
- The plumber worked for 35 hours and the cost for his work is not mentioned.
- The electrician worked for 42 hours and the cost for his work is not mentioned.
- The painter worked for 29 hours and the cost for his work is not mentioned.
However, the total cost for the three workers combined is given as $2,570.
So, let's assume the plumber's hourly rate is "p" dollars per hour, the electrician's hourly rate is "e" dollars per hour, and the painter's hourly rate is "r" dollars per hour. Given this information, we can set up the following equations:
35p + 42e + 29r = 2570 (Equation 1)
Similarly, for the Watson home:
28p + 28e + 28r = 1960 (Equation 2)
And for the Miller home:
24e + 30p + 25r = 1845 (Equation 3)
We now have three equations with three unknowns. We can solve this system of equations to find the values of p, e, and r.
To solve this system, you have a few options:
- You can use elimination: Multiply one equation or both equations to get a variable to cancel out, then solve for the remaining variables.
- You can use substitution: Solve one equation for one variable and substitute it into the other equations.
- You can use matrix methods or software to solve the system.
Once you solve the system of equations, you will have the values of p, e, and r, which represent the hourly rate for the plumber, electrician, and painter, respectively.