Solve {l}{2x_{1}+2x_{2}-x_{3}=0}{x_{1}-2x_{2}+4x_{3}=0}{5x_{1}+7x_{2}+x_{3}=0}

$$\left\{ \begin{array} { l } { 2 x _ { 1 } + 2 x _ { 2 } - x _ { 3 } = 0 } \\ { x _ { 1 } - 2 x _ { 2 } + 4 x _ { 3 } = 0 } \\ { 5 x _ { 1 } + 7 x _ { 2 } + x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=0$$

$$x_{2}=0$$

$$x_{3}=0$$

Short Steps Using Substitution

Solve 2x_{1}+2x_{2}-x_{3}=0 for x_{3}.

$$x_{3}=2x_{1}+2x_{2}$$

Substitute 2x_{1}+2x_{2} for x_{3} in the second and third equation.

$$x_{1}-2x_{2}+4\left(2x_{1}+2x_{2}\right)=0 5x_{1}+7x_{2}+2x_{1}+2x_{2}=0$$

Solve these equations for x_{2} and x_{1} respectively.

$$x_{2}=-\frac{3}{2}x_{1} x_{1}=-\frac{9}{7}x_{2}$$

Substitute -\frac{3}{2}x_{1} for x_{2} in the equation x_{1}=-\frac{9}{7}x_{2}.

$$x_{1}=-\frac{9}{7}\left(-\frac{3}{2}\right)x_{1}$$

Solve x_{1}=-\frac{9}{7}\left(-\frac{3}{2}\right)x_{1} for x_{1}.

$$x_{1}=0$$

Substitute 0 for x_{1} in the equation x_{2}=-\frac{3}{2}x_{1}.

$$x_{2}=-\frac{3}{2}\times 0$$

Calculate x_{2} from x_{2}=-\frac{3}{2}\times 0.

$$x_{2}=0$$

Substitute 0 for x_{2} and 0 for x_{1} in the equation x_{3}=2x_{1}+2x_{2}.

$$x_{3}=2\times 0+2\times 0$$

Calculate x_{3} from x_{3}=2\times 0+2\times 0.

$$x_{3}=0$$

The system is now solved.

$$x_{1}=0 x_{2}=0 x_{3}=0$$