Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION.) left{begin{matrix} sqrt{2} x + y + 2z = 0 \ sqrt{2}y – 3z = –

Solution using Gauss-Jordan elimination: We can write the system of equations in the augmented matrix form: $$left[begin{array}{@{}ccc|c@{}} sqrt{2} & 1 & 2 & 0 \ 0 & sqrt{2} & -3 & -2sqrt{2} \ 0 & -1 & sqrt{2} & 2 end{array}right]$$ We can begin by using row operations to try to eliminate the off-diagonal entries in the first column. First, we can subtract $frac{1}{sqrt{2}}$ times the first row from the second row to get: $$left[begin{array}{@{}ccc|c@{}} sqrt{2} & 1 & 2 & 0 \ 0 & 0 & -frac{7}{sqrt{2}} & -2 \ 0 & -1 & sqrt{2} & 2 end{array}right]$$ Next, we can swap the second and third rows and multiply the second row by $-sqrt{2}$: $$left[begin{array}{@{}ccc|c@{}} sqrt{2} & 1 & 2 & 0 \ 0 & 1 & -sqrt{2} & sqrt{2} \ 0 & 0 & -frac{7}{sqrt{2}} & -2 end{array}right]$$ Finally, we can multiply the last row by $-frac{sqrt{2}}{7}$ to get: $$left[begin{array}{@{}ccc|c@{}} sqrt{2} & 1 & 2 & 0 \ 0 & 1 & -sqrt{2} & sqrt{2} \ 0 & 0 & 1 & \frac{2sqrt{2}}{7} end{array}right]$$ Now, we can use row operations to eliminate the off-diagonal entries in the second column. First, we can subtract $2$ times the third row from the first row and add $sqrt{2}$ times the third row to the second row: $$left[begin{array}{@{}ccc|c@{}} sqrt{2} & 1 & 0 & -frac{4sqrt{2}}{7} \ 0 & 1 & 0 & \frac{4sqrt{2}}{7} \ 0 & 0 & 1 & \frac{2sqrt{2}}{7} end{array}right]$$ Finally, we can subtract the second row from the first row to get: $${left[begin{array}{@{}ccc|c@{}} sqrt{2} & 0 & 0 & 0 \ 0 & 1 & 0 & \frac{4sqrt{2}}{7} \ 0 & 0 & 1 & \frac{2sqrt{2}}{7} end{array}right]}$$ Thus, the solution is $x=0$, $y=frac{4sqrt{2}}{7}$, $z=frac{2sqrt{2}}{7}$.