Question

If a building contractor hires 7 day laborers and 2 concrete finishers​, his payroll for the day is ​$1328. If he hires 1 day laborer and 4 concrete​ finishers, his daily cost is ​$784. Find the daily wage for each type of worker. Solve the problem using matrices.

Answer 1

The daily wage for a day laborer is $144, and for a concrete finisher, it is $160. This is determined by solving a system of linear equations using matrix multiplication.

Let's denote the daily wage for a day laborer as x and for a concrete finisher as y. The information given can be expressed in a system of linear equations:

1. For the scenario with 7 day laborers and 2 concrete finishers: 7x + 2y = 1328

2. For the scenario with 1 day laborer and 4 concrete finishers: x + 4y = 784

We can represent this system of equations in matrix form Ax = B, where:

`\[ A = \begin{bmatrix} 7 & 2 \\ 1 & 4 \end{bmatrix} \]\[ x = \begin{bmatrix} x \\ y \end{bmatrix} \]\[ B = \begin{bmatrix} 1328 \\ 784 \end{bmatrix} \]`

Now, to solve for x, multiply both sides by the inverse of A:

`\[ A^(-1)Ax = A^(-1)B \]\[ x = A^(-1)B \]`

Let's calculate the inverse of matrix A and then find x.

The inverse of matrix A is given by:

`\[ A^(-1) = (1)/((7 * 4 - 1 * 2)) \begin{bmatrix} 4 & -2 \\ -1 & 7 \end{bmatrix} \]\[ A^(-1) = (1)/(26) \begin{bmatrix} 4 & -2 \\ -1 & 7 \end{bmatrix} \]`

Now, multiply
`\(A^(-1)\)` by B to find x:

`\[ x = A^(-1)B \]\[ x = (1)/(26) \begin{bmatrix} 4 & -2 \\ -1 & 7 \end{bmatrix} \begin{bmatrix} 1328 \\ 784 \end{bmatrix} \]`

After performing the matrix multiplication, you'll get the values for x and y, representing the daily wage for a day laborer and a concrete finisher, respectively.

Let's continue with the calculations:

`\[ x = (1)/(26) \begin{bmatrix} 4 & -2 \\ -1 & 7 \end{bmatrix} \begin{bmatrix} 1328 \\ 784 \end{bmatrix} \]\[ x = (1)/(26) \begin{bmatrix} (4 * 1328) + (-2 * 784) \\ (-1 * 1328) + (7 * 784) \end{bmatrix} \]\[ x = (1)/(26) \begin{bmatrix} 5312 - 1568 \\ -1328 + 5488 \end{bmatrix} \]\[ x = (1)/(26) \begin{bmatrix} 3744 \\ 4160 \end{bmatrix} \]\[ x = \begin{bmatrix} 144 \\ 160 \end{bmatrix} \]`

Therefore, the daily wage for a day laborer (x) is $144, and for a concrete finisher (y), it is $160.