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

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

Solve for x_1, x_2, x_3

$$x_{1}=0$$

$$x_{2}=0$$

$$x_{3}=0$$

Short Steps Using Substitution

Solve 3x_{1}+5x_{2}-x_{3}=0 for x_{3}.

$$x_{3}=3x_{1}+5x_{2}$$

Substitute 3x_{1}+5x_{2} for x_{3} in the second and third equation.

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

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

$$x_{2}=-\frac{7}{11}x_{1} x_{1}=-\frac{1}{2}x_{2}$$

Substitute -\frac{7}{11}x_{1} for x_{2} in the equation x_{1}=-\frac{1}{2}x_{2}.

$$x_{1}=-\frac{1}{2}\left(-\frac{7}{11}\right)x_{1}$$

Solve x_{1}=-\frac{1}{2}\left(-\frac{7}{11}\right)x_{1} for x_{1}.

$$x_{1}=0$$

Substitute 0 for x_{1} in the equation x_{2}=-\frac{7}{11}x_{1}.

$$x_{2}=-\frac{7}{11}\times 0$$

Calculate x_{2} from x_{2}=-\frac{7}{11}\times 0.

$$x_{2}=0$$

Substitute 0 for x_{2} and 0 for x_{1} in the equation x_{3}=3x_{1}+5x_{2}.

$$x_{3}=3\times 0+5\times 0$$

Calculate x_{3} from x_{3}=3\times 0+5\times 0.

$$x_{3}=0$$

The system is now solved.

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