# How do you solve using gaussian elimination or gauss-jordan elimination, 2x + 5y - 2z = 14, 5x -6y + 2z = 0, 4x - y + 3z = -7?

Jan 6, 2018

$P = \left\{\left(2 , 0 , - 5\right)\right\}$

#### Explanation:

$\left(\begin{matrix}2 & 5 & - 2 & | & 14 \\ 5 & - 6 & 2 & | & 0 \\ 4 & - 1 & 3 & | & - 7\end{matrix}\right) \approx \left(\begin{matrix}10 & 25 & - 10 & | & 70 \\ 10 & - 12 & 4 & | & 0 \\ 0 & - 11 & 7 & | & - 35\end{matrix}\right)$
${R}_{3} = {R}_{3} - 2 \times {R}_{1}$
${R}_{2} = {R}_{2} \times 2$
${R}_{1} = {R}_{1} \times 5$
${R}_{2} = {R}_{2} - {R}_{1}$

$\left(\begin{matrix}10 & 25 & - 10 & | & 70 \\ 0 & - 37 & 14 & | & - 70 \\ 0 & - 11 & 7 & | & - 35\end{matrix}\right) \approx \left(\begin{matrix}10 & 25 & - 10 & | & 70 \\ 0 & - 407 & 154 & | & - 770 \\ 0 & - 407 & 259 & | & - 1295\end{matrix}\right)$
${R}_{2} = {R}_{2} \times 11$
${R}_{3} = {R}_{3} \times 37$
${R}_{3} = {R}_{3} - {R}_{2}$

$\left(\begin{matrix}10 & 25 & - 10 & | & 70 \\ 0 & - 407 & 154 & | & - 770 \\ 0 & 0 & 105 & | & - 525\end{matrix}\right) \approx \left(\begin{matrix}10 & 25 & - 10 & | & 70 \\ 0 & - 37 & 14 & | & - 70 \\ 0 & 0 & 1 & | & - 5\end{matrix}\right)$
${R}_{3} = {R}_{3} \times \frac{1}{105}$
${R}_{2} = {R}_{2} \times \frac{1}{11}$
${R}_{2} = {R}_{2} - 14 \times {R}_{3}$
${R}_{1} = {R}_{1} + 10 \times {R}_{3}$

$\left(\begin{matrix}10 & 25 & 0 & | & 20 \\ 0 & - 37 & 0 & | & 0 \\ 0 & 0 & 1 & | & - 5\end{matrix}\right)$
${R}_{2} = {R}_{2} \times - \frac{1}{37}$
${R}_{1} = {R}_{1} - 25 \times {R}_{2}$
${R}_{1} = {R}_{1} \times \frac{1}{10}$

$\left(\begin{matrix}1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 0 \\ 0 & 0 & 1 & | & - 5\end{matrix}\right)$

$P = \left\{\left(2 , 0 , - 5\right)\right\}$

Jan 7, 2018

$x = 2$, $y = 0$ and $z = - 5$

#### Explanation:

From third equation, $y = 4 x + 3 z + 7$

Hence,

$2 x + 5 \cdot \left(4 x + 3 z + 7\right) - 2 z = 14$ or $22 x + 13 z = - 21$ $\left(1\right)$ and,

$5 x - 6 \cdot \left(4 x + 3 z + 7\right) + 2 z = 0$ or $- 19 x - 16 z = 42$ $\left(2\right)$

From $\left(1\right)$, $z = \frac{- 22 x - 21}{13}$

Consequently,

$- 19 x - 16 \cdot \frac{- 22 x - 21}{13} = 42$

$\frac{352 x + 336}{13} - 19 x = 42$

$\frac{352 x + 336}{13} = 19 x + 42$

$352 x + 336 = 13 \cdot \left(19 x + 42\right)$

$352 x + 336 = 247 x + 546$

$105 x = 210$, so $x = 2$

Thus,

$z = \frac{\left(- 22\right) \cdot 2 - 21}{13} = - 5$ and,

$y = 4 \cdot 2 + 3 \cdot \left(- 5\right) + 7 = 0$