Connor Randall

2022-01-20

Two welders worked a total of 47 h on a project. One welder made $37/h, while the other made$39/h. If the gross earnings of the two welders was $1781 for the job, how many hours did each welder work? Using row- echron matrix. Answer & Explanation Deegan Mullen Expert 2022-01-21Added 12 answers Step 1 Solving a system of equations using row-echlon form can be done by reducing the given matrix to another matrix, using transformations, such that each entry below the diagonal element of each column is 0. Step 2 Let the two welders work for x and y hours respectively. As the total hours for which the two welders work is 47, hence we will have: $x+y=47$ (i) Also, the welders earn at a rate of 37 h and 39 h respectively and their gross earning was$1781. Hence we will have:
$37x+39y=1781$ (ii)
Thus we have a system of two linear equations which can be transformed to the matrix form as shown below:
$\left[\begin{array}{cc}1& 1\\ 37& 38\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 1781\end{array}\right]$
We can solve this system of equations by transforming it into a row-echlon form as shown below:
$\left[\begin{array}{cc}1& 1\\ 37& 39\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 1781\end{array}\right]$
${R}_{2}\to {R}_{2}-37{R}_{1}$
$\left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 42\end{array}\right]$
Note: Here we already obtain the row−echlon form and can use it to find the required values of x and y, but we go a step further and completelpy convert the matrix into an identity matrix to directly obtain the required values.
${R}_{2}\to \frac{1}{2}{R}_{2}$
$\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 21\end{array}\right]$
${R}_{1}\to {R}_{1}-{R}_{2}$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}26\\ 21\end{array}\right]$
Thus we get the solution as, $x=26$ and $y=21$. Hence, the welders work for 26 hrs and 21 hrs respectively.

Do you have a similar question?