Solve {l}{2x_{1}-x_{2}-x_{4}=0}{x_{1}+3x_{2}+3x_{3}+5x_{4}=0}{2x_{2}+4x_{3}+6x_{4}=0}

$$\left\{ \begin{array} { l } { 2 x _ { 1 } - x _ { 2 } - x _ { 4 } = 0 } \\ { x _ { 1 } + 3 x _ { 2 } + 3 x _ { 3 } + 5 x _ { 4 } = 0 } \\ { 2 x _ { 2 } + 4 x _ { 3 } + 6 x _ { 4 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{x_{4}}{4}$$

$$x_{2}=-\frac{x_{4}}{2}$$

$$x_{3}=-\frac{5x_{4}}{4}$$

Short Steps Using Substitution

Solve 2x_{1}-x_{2}-x_{4}=0 for x_{2}.

$$x_{2}=2x_{1}-x_{4}$$

Substitute 2x_{1}-x_{4} for x_{2} in the second and third equation.

$$x_{1}+3\left(2x_{1}-x_{4}\right)+3x_{3}+5x_{4}=0 2\left(2x_{1}-x_{4}\right)+4x_{3}+6x_{4}=0$$

Solve these equations for x_{1} and x_{3} respectively.

$$x_{1}=-\frac{3}{7}x_{3}-\frac{2}{7}x_{4} x_{3}=-x_{1}-x_{4}$$

Substitute -\frac{3}{7}x_{3}-\frac{2}{7}x_{4} for x_{1} in the equation x_{3}=-x_{1}-x_{4}.

$$x_{3}=-\left(-\frac{3}{7}x_{3}-\frac{2}{7}x_{4}\right)-x_{4}$$

Solve x_{3}=-\left(-\frac{3}{7}x_{3}-\frac{2}{7}x_{4}\right)-x_{4} for x_{3}.

$$x_{3}=-\frac{5}{4}x_{4}$$

Substitute -\frac{5}{4}x_{4} for x_{3} in the equation x_{1}=-\frac{3}{7}x_{3}-\frac{2}{7}x_{4}.

$$x_{1}=-\frac{3}{7}\left(-\frac{5}{4}\right)x_{4}-\frac{2}{7}x_{4}$$

Calculate x_{1} from x_{1}=-\frac{3}{7}\left(-\frac{5}{4}\right)x_{4}-\frac{2}{7}x_{4}.

$$x_{1}=\frac{1}{4}x_{4}$$

Substitute \frac{1}{4}x_{4} for x_{1} in the equation x_{2}=2x_{1}-x_{4}.

$$x_{2}=2\times \frac{1}{4}x_{4}-x_{4}$$

Calculate x_{2} from x_{2}=2\times \frac{1}{4}x_{4}-x_{4}.

$$x_{2}=-\frac{1}{2}x_{4}$$

The system is now solved.

$$x_{1}=\frac{1}{4}x_{4} x_{2}=-\frac{1}{2}x_{4} x_{3}=-\frac{5}{4}x_{4}$$