If a building contractor hires 5 day laborers and 2 concrete finishers, his payroll for the day is $1285. If he hires 1 day laborer and 5 concrete finishers, his daily cost is $1177. Find the daily wage for each type of worker. Solve the problem using matrices.
you need to solve the system
5d+2c = 1285
1d+5c = 1177
So, you have AX = B,
(5 2) (d) = (1285)
(1 5) (c) = (1177)
A-1 =
(5/23 -2/23)
(-1/23 5/23)
X = A-1 * B, so
(d) = 200
(c) = 177
To solve the problem using matrices, let's assign variables to the daily wages of the day laborers and concrete finishers.
Let's denote the daily wage for a day laborer as 'x' and the daily wage for a concrete finisher as 'y'.
Now we can set up a system of linear equations based on the given information:
Equation 1: 5x + 2y = 1285
Equation 2: 1x + 5y = 1177
We can rewrite these equations in matrix form as follows:
| 5 2 | | x | | 1285 |
| 1 5 | | y | = | 1177 |
To solve this system of equations, we can use matrix inversion. First, let's find the inverse of the coefficient matrix:
| 5 2 |
| 1 5 |
The inverse of this matrix is:
1/23 * | 5 -2 |
| -1 5 |
Now, we can multiply the inverse matrix by the column matrix of the constants:
1/23 * ( | 5 -2 | ) * ( | 1285 | ) = ( | x | )
( | -1 5 | ) ( | 1177 | ) ( | y | )
Simplifying this equation, we get:
( 5x - 2y ) / 23 = 1285/23
( -x + 5y ) / 23 = 1177/23
Now, we can solve these equations individually to find the values of x and y.
Equation 1: 5x - 2y = 1285/23
5x = (1285/23) + (2y)
x = (1/5) * ((1285/23) + (2y))
Equation 2: -x + 5y = 1177/23
5y = (1177/23) + x
y = (1/5) * ((1177/23) + x)
Hence, the daily wage for the day laborers (x) is (1/5) * ((1285/23) + (2y)), and the daily wage for the concrete finishers (y) is (1/5) * ((1177/23) + x).
To find the exact values for x and y, you need to substitute these equations back into either of the original equations.