Two welders worked a total of 47 h on a project. One welder made $37/h,...

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:
${\displaystyle x+y=47}$ (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:
${\displaystyle 37x+39y=1781}$ (ii)
Thus we have a system of two linear equations which can be transformed to the matrix form as shown below:
${\displaystyle \left[\begin{array}{cc}1& 1\\ 37& 38\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 1781\end{array}\right]}$
We can solve this system of equations by transforming it into a row-echlon form as shown below:
${\displaystyle \left[\begin{array}{cc}1& 1\\ 37& 39\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 1781\end{array}\right]}$
${\displaystyle R_2\to R_2-37R_1}$
${\displaystyle \left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 42\end{array}\right]}$
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.
${\displaystyle R_2\to \frac{1}{2}{R}_2}$
${\displaystyle \left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}47\\ 21\end{array}\right]}$
${\displaystyle R_1\to R_1-R_2}$
${\displaystyle \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{c}26\\ 21\end{array}\right]}$
Thus we get the solution as, ${\displaystyle x=26}$ and ${\displaystyle y=21}$. Hence, the welders work for 26 hrs and 21 hrs respectively.