Solve {l}{4x_{1}+2x_{2}-x_{3}=2}{3x_{1}-x_{2}+2x_{3}=10}{11x_{1}+x_{2}=8}

$$\left\{ \begin{array} { l } { 4 x _ { 1 } + 2 x _ { 2 } - x _ { 3 } = 2 } \\ { 3 x _ { 1 } - x _ { 2 } + 2 x _ { 3 } = 10 } \\ { 11 x _ { 1 } + x _ { 2 } = 8 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{5}{11}\approx 0.454545455$$

$$x_{2}=3$$

$$x_{3} = \frac{64}{11} = 5\frac{9}{11} \approx 5.818181818$$

Short Steps Using Substitution

Solve 4x_{1}+2x_{2}-x_{3}=2 for x_{3}.

$$x_{3}=-2+4x_{1}+2x_{2}$$

Substitute -2+4x_{1}+2x_{2} for x_{3} in the equation 3x_{1}-x_{2}+2x_{3}=10.

$$3x_{1}-x_{2}+2\left(-2+4x_{1}+2x_{2}\right)=10$$

Solve the second equation for x_{2} and the third equation for x_{1}.

$$x_{2}=-\frac{11}{3}x_{1}+\frac{14}{3} x_{1}=-\frac{1}{11}x_{2}+\frac{8}{11}$$

Substitute -\frac{11}{3}x_{1}+\frac{14}{3} for x_{2} in the equation x_{1}=-\frac{1}{11}x_{2}+\frac{8}{11}.

$$x_{1}=-\frac{1}{11}\left(-\frac{11}{3}x_{1}+\frac{14}{3}\right)+\frac{8}{11}$$

Solve x_{1}=-\frac{1}{11}\left(-\frac{11}{3}x_{1}+\frac{14}{3}\right)+\frac{8}{11} for x_{1}.

$$x_{1}=\frac{5}{11}$$

Substitute \frac{5}{11} for x_{1} in the equation x_{2}=-\frac{11}{3}x_{1}+\frac{14}{3}.

$$x_{2}=-\frac{11}{3}\times \frac{5}{11}+\frac{14}{3}$$

Calculate x_{2} from x_{2}=-\frac{11}{3}\times \frac{5}{11}+\frac{14}{3}.

$$x_{2}=3$$

Substitute 3 for x_{2} and \frac{5}{11} for x_{1} in the equation x_{3}=-2+4x_{1}+2x_{2}.

$$x_{3}=-2+4\times \frac{5}{11}+2\times 3$$

Calculate x_{3} from x_{3}=-2+4\times \frac{5}{11}+2\times 3.

$$x_{3}=\frac{64}{11}$$

The system is now solved.

$$x_{1}=\frac{5}{11} x_{2}=3 x_{3}=\frac{64}{11}$$