Solve {l}{x_{1}+2x_{2}+3x_{3}=6}{2x_{1}+3x_{2}-4x_{3}=1}{7x_{1}-2x_{2}-5x_{3}=0}

$$\left\{ \begin{array}{l}{ x _ { 1 } + 2 x _ { 2 } + 3 x _ { 3 } = 6 }\\{ 2 x _ { 1 } + 3 x _ { 2 } - 4 x _ { 3 } = 1 }\\{ 7 x _ { 1 } - 2 x _ { 2 } - 5 x _ { 3 } = 0 }\end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=1$$

$$x_{2}=1$$

$$x_{3}=1$$

Short Steps Using Substitution

Solve x_{1}+2x_{2}+3x_{3}=6 for x_{1}.

$$x_{1}=-2x_{2}-3x_{3}+6$$

Substitute -2x_{2}-3x_{3}+6 for x_{1} in the second and third equation.

$$2\left(-2x_{2}-3x_{3}+6\right)+3x_{2}-4x_{3}=1 7\left(-2x_{2}-3x_{3}+6\right)-2x_{2}-5x_{3}=0$$

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

$$x_{2}=-10x_{3}+11 x_{3}=\frac{21}{13}-\frac{8}{13}x_{2}$$

Substitute -10x_{3}+11 for x_{2} in the equation x_{3}=\frac{21}{13}-\frac{8}{13}x_{2}.

$$x_{3}=\frac{21}{13}-\frac{8}{13}\left(-10x_{3}+11\right)$$

Solve x_{3}=\frac{21}{13}-\frac{8}{13}\left(-10x_{3}+11\right) for x_{3}.

$$x_{3}=1$$

Substitute 1 for x_{3} in the equation x_{2}=-10x_{3}+11.

$$x_{2}=-10+11$$

Calculate x_{2} from x_{2}=-10+11.

$$x_{2}=1$$

Substitute 1 for x_{2} and 1 for x_{3} in the equation x_{1}=-2x_{2}-3x_{3}+6.

$$x_{1}=-2-3+6$$

Calculate x_{1} from x_{1}=-2-3+6.

$$x_{1}=1$$

The system is now solved.

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