# How do you solve using gaussian elimination or gauss-jordan elimination, x_3 + x_4 = 0, x_1 + x_2 + x_3 + x_4 = 1, 2x_1 - x_2 + x_3 + 2x_4 = 0, 2x_1 - x_2 + x_3 + x_4 = 0?

Dec 10, 2017

${x}_{1} = \frac{1}{3}$, ${x}_{2} = \frac{2}{3}$, ${x}_{3} = {x}_{4} = 0$

#### Explanation:

From first equation, ${x}_{4} = - {x}_{3}$

After using this equality into other ones,

${x}_{1} + {x}_{2} = 1$ $\left(a\right)$, $2 {x}_{1} - {x}_{2} - {x}_{3} = 0$ $\left(b\right)$ and $2 {x}_{1} - {x}_{2} = 0$ $\left(c\right)$

From $\left(c\right)$, ${x}_{2} = 2 {x}_{1}$

After using this equality into other ones,

${x}_{1} + 2 {x}_{1} = 1$ or $3 {x}_{1} = 1$ $\left(d\right)$ and $2 {x}_{1} - 2 {x}_{1} - {x}_{3} = 0$ or $- {x}_{3} = 0$ $\left(e\right)$

From $\left(d\right)$, ${x}_{1} = \frac{1}{3}$ and $\left(e\right)$, ${x}_{3} = 0$

Hence, ${x}_{2} = \frac{2}{3}$ from $\left(c\right)$ and ${x}_{4} = 0$ from first equation.