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

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

Solve for x_1, x_2, x_3

$$x_{1}=2$$

$$x_{2}=-1$$

$$x_{3}=1$$

Short Steps Using Substitution

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

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

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

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

Solve the second equation for x_{2} and the third equation for x_{3}.

$$x_{2}=-\frac{7}{2}+\frac{5}{2}x_{3} x_{3}=-x_{2}$$

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

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

Solve x_{3}=-\left(-\frac{7}{2}+\frac{5}{2}x_{3}\right) for x_{3}.

$$x_{3}=1$$

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

$$x_{2}=-\frac{7}{2}+\frac{5}{2}\times 1$$

Calculate x_{2} from x_{2}=-\frac{7}{2}+\frac{5}{2}\times 1.

$$x_{2}=-1$$

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

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

Calculate x_{1} from x_{1}=3\left(-1\right)+5.

$$x_{1}=2$$

The system is now solved.

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