set up the augmented matrix.\n2x + 4y - 10z = 40\n3x + 5y - 11z = 56\nx + y - z = 16\nwe have chosen to use…

Explanation:

Step1: Recall row - operation rule

To perform $R_1+R_2\rightarrow R_2$, we add the corresponding elements of row 1 and row 2 and put the result in row 2. For $R_1 + R_3\rightarrow R_3$, we add the corresponding elements of row 1 and row 3 and put the result in row 3.

Step2: Calculate $R_1+R_2\rightarrow R_2$

The first row $R_1=\left[1\ 1\ - 1\ 16\right]$ and the second row $R_2=\left[3\ 5\ -11\ 56\right]$. Adding them element - by - element: [ \begin{align*} 1 + 3&=4\ 1+5&=6\ -1+( - 11)&=-12\ 16 + 56&=72 \end{align*} ] So the new second row is $\left[4\ 6\ -12\ 72\right]$.

Step3: Calculate $R_1+R_3\rightarrow R_3$

The first row $R_1=\left[1\ 1\ - 1\ 16\right]$ and the third row $R_3=\left[2\ 4\ -10\ 40\right]$. Adding them element - by - element: [ \begin{align*} 1+2&=3\ 1 + 4&=5\ -1+( - 10)&=-11\ 16+40&=56 \end{align*} ] So the new third row is $\left[3\ 5\ -11\ 56\right]$.

Answer:

For $R_1 + R_2\rightarrow R_2$: $\left[4\ 6\ -12\ 72\right]$ For $R_1 + R_3\rightarrow R_3$: $\left[3\ 5\ -11\ 56\right]$