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

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

Solve for x_1, x_2, x_3

$$x_{1}=1$$

$$x_{2}=0$$

$$x_{3}=0$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x_{2} and x_{3} respectively.

$$x_{2}=-\frac{1}{2}x_{3} x_{3}=-\frac{1}{8}x_{2}$$

Substitute -\frac{1}{2}x_{3} for x_{2} in the equation x_{3}=-\frac{1}{8}x_{2}.

$$x_{3}=-\frac{1}{8}\left(-\frac{1}{2}\right)x_{3}$$

Solve x_{3}=-\frac{1}{8}\left(-\frac{1}{2}\right)x_{3} for x_{3}.

$$x_{3}=0$$

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

$$x_{2}=-\frac{1}{2}\times 0$$

Calculate x_{2} from x_{2}=-\frac{1}{2}\times 0.

$$x_{2}=0$$

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

$$x_{1}=-2\times 0-3\times 0+1$$

Calculate x_{1} from x_{1}=-2\times 0-3\times 0+1.

$$x_{1}=1$$

The system is now solved.

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