Question

Write a system of linear equations for the problem and solve the system using an augmented matrix and your calculator.

Mike works a total of 59hr per week at his two jobs. He makes ​$8 per hour at job A and ​$9 per hour at job B. If his total pay for one week is ​$494 before​ taxes, then how many hours does he work at each​ job?

Answer 1

Answer: he works 37 hours in job A, and 22 hours in job B.

Explanation:

Let's define the variables:

A = number of hours that he works at Job A

B = number of hours that he works at Job B.

We know that he works a total of 59hr per week, then we have:

A + B = 59

And we also know that he makes $494 in one week, then we have the equation:

A*$8 + B*$9 = $494.

Then we have the system of equations:

A + B = 59

A*$8 + B*$9 = $494

To solve this with an augmented matrix, we need to turn these equations into a matrix, where we only write the coefficients as follows:

`\left[\begin{array}{cccc}1&1&I&59\\8&9&I&494\end{array}\right]`

(The I's should represent a line for the augmented matrix, that line separates the values of the coefficients to the right value of each equation)

The first column represents the A coefficients, the second column represents the B coefficients.

To solve this, we want to get something of the form:

`\left[\begin{array}{cccc}1&0&I&x\\0&1&I&y\end{array}\right]`

Where x and y are our solutions.

To do it we can just do: Column 2 - 8*column 1, this will get us to:

`\left[\begin{array}{cccc}1&1&I&59\\8 - 8*1&9 - 8*1&I&494 - 8*59\end{array}\right]`

=

`\left[\begin{array}{cccc}1&1&I&59\\0&1&I&22\end{array}\right]`

Now we can do: Column1 - column2, to get

`\left[\begin{array}{cccc}1 - 0&1 - 1&I&59 - 22\\0&1&I&22\end{array}\right]`

`\left[\begin{array}{cccc}1&0&I&37\\0&1&I&22\end{array}\right]`

Then the solutions are:

1*A =37

1*B = 22

This means that he works 37 hours in job A, and 22 hours in job B.