# How do you solve using gaussian elimination or gauss-jordan elimination, 10x-7y+3z+5u=6, -6x+8y-z-4u=5, 3x+y+4z+11u=2, 5x-9y-2z+4u=7?

Jan 7, 2018

$x = \frac{22 + 7 \times \left(\frac{44}{19}\right)}{10} = 3.8210526315789473684210526315789$

$y = \frac{44}{19} = 2.3157894736842105263157894736842$

$z = - 7$

$u = 1$

#### Explanation:

$\left(\begin{matrix}10 & - 7 & 3 & 5 & | & 6 \\ - 6 & 8 & - 1 & - 4 & | & 5 \\ 3 & 1 & 4 & 11 & | & 2 \\ 5 & - 9 & - 2 & 4 & | & 7\end{matrix}\right) \approx \left(\begin{matrix}10 & - 7 & 3 & 5 & | & 6 \\ - 6 & 8 & - 1 & - 4 & | & 5 \\ 6 & 2 & 8 & 22 & | & 4 \\ 10 & - 18 & - 4 & 8 & | & 14\end{matrix}\right)$
${R}_{4} = {R}_{4} \times 2$
${R}_{3} = {R}_{3} \times 2$
${R}_{4} = {R}_{4} - {R}_{1}$
${R}_{3} = {R}_{3} + {R}_{2}$

$\left(\begin{matrix}10 & - 7 & 3 & 5 & | & 6 \\ - 6 & 8 & - 1 & - 4 & | & 5 \\ 0 & 10 & 7 & 18 & | & 9 \\ 0 & - 11 & - 7 & 3 & | & 8\end{matrix}\right) \approx \left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ - 30 & 40 & - 5 & - 20 & | & 25 \\ 0 & 10 & 7 & 18 & | & 9 \\ 0 & - 11 & - 7 & 3 & | & 8\end{matrix}\right)$
${R}_{1} = {R}_{1} \times 3$
${R}_{2} = {R}_{2} \times 5$
${R}_{2} = {R}_{2} + {R}_{1}$

$\left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 19 & 4 & - 5 & | & 43 \\ 0 & 10 & 7 & 18 & | & 9 \\ 0 & - 11 & - 7 & 3 & | & 8\end{matrix}\right) \approx \left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 19 & 4 & - 5 & | & 43 \\ 0 & 10 & 7 & 18 & | & 9 \\ 0 & - 110 & - 70 & 30 & | & 80\end{matrix}\right)$
${R}_{4} = {R}_{4} \cdot 10$
${R}_{4} = {R}_{4} + 11 \times {R}_{3}$

$\left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 19 & 4 & - 5 & | & 43 \\ 0 & 10 & 7 & 18 & | & 9 \\ 0 & 0 & 7 & 228 & | & 179\end{matrix}\right) \approx \left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 190 & 40 & - 50 & | & 430 \\ 0 & 190 & 133 & 342 & | & 171 \\ 0 & 0 & 7 & 228 & | & 179\end{matrix}\right)$
${R}_{3} = {R}_{3} \times 19$
${R}_{2} = {R}_{2} \times 10$
${R}_{3} = {R}_{3} - {R}_{2}$

$\left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 190 & 40 & - 50 & | & 430 \\ 0 & 0 & 93 & 392 & | & - 259 \\ 0 & 0 & 7 & 228 & | & 179\end{matrix}\right) \approx \left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 190 & 40 & - 50 & | & 430 \\ 0 & 0 & 651 & 2744 & | & - 1813 \\ 0 & 0 & 651 & 21204 & | & 16647\end{matrix}\right)$
${R}_{4} = {R}_{4} \times 93$
${R}_{3} = {R}_{3} \times 7$
${R}_{4} = {R}_{4} - {R}_{3}$

$\left(\begin{matrix}30 & - 21 & 9 & 15 & | & 18 \\ 0 & 190 & 40 & - 50 & | & 430 \\ 0 & 0 & 651 & 2744 & | & - 1813 \\ 0 & 0 & 0 & 18460 & | & 18460\end{matrix}\right) \approx \left(\begin{matrix}10 & - 7 & 3 & 5 & | & 6 \\ 0 & 19 & 4 & - 5 & | & 43 \\ 0 & 0 & 93 & 392 & | & - 259 \\ 0 & 0 & 0 & 1 & | & 1\end{matrix}\right)$
Dividing what is possible
${R}_{3} = {R}_{3} - 392 \times {R}_{4}$
${R}_{2} = {R}_{2} + 5 \times {R}_{4}$
${R}_{2} = {R}_{2} - 5 \times {R}_{4}$

$\left(\begin{matrix}10 & - 7 & 3 & 0 & | & 1 \\ 0 & 19 & 4 & 0 & | & 48 \\ 0 & 0 & 93 & 0 & | & - 651 \\ 0 & 0 & 0 & 1 & | & 1\end{matrix}\right) \approx \left(\begin{matrix}10 & - 7 & 0 & 0 & | & 22 \\ 0 & 19 & 0 & 0 & | & 44 \\ 0 & 0 & 1 & 0 & | & - 7 \\ 0 & 0 & 0 & 1 & | & 1\end{matrix}\right)$
${R}_{3} = {R}_{3} \times \frac{1}{93}$
${R}_{2} = {R}_{2} - 4 \times {R}_{3}$
${R}_{1} = {R}_{1} - 3 \times {R}_{3}$
${R}_{2} = {R}_{2} \times \frac{1}{19}$
${R}_{1} = {R}_{1} + 7 \times {R}_{2}$

$\left(\begin{matrix}1 & 0 & 0 & 0 & | & \frac{22 + 7 \times \left(\frac{44}{19}\right)}{10} \\ 0 & 1 & 0 & 0 & | & \frac{44}{19} \\ 0 & 0 & 1 & 0 & | & - 7 \\ 0 & 0 & 0 & 1 & | & 1\end{matrix}\right)$

$x = \frac{22 + 7 \times \left(\frac{44}{19}\right)}{10} = 3.8210526315789473684210526315789$

$y = \frac{44}{19} = 2.3157894736842105263157894736842$

$z = - 7$

$u = 1$