Solve {c}{2x_{1}-x_{2}+3x_{3}=1}{4x_{1}+2x_{2}+5x_{3}=4}{x_{1}+2x_{2}=7}

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

Solve for x_1, x_2, x_3

$$x_{1}=9$$

$$x_{2}=-1$$

$$x_{3}=-6$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x_{1} and x_{3} respectively.

$$x_{1}=-\frac{11}{8}x_{3}+\frac{3}{4} x_{3}=\frac{3}{2}-\frac{5}{6}x_{1}$$

Substitute -\frac{11}{8}x_{3}+\frac{3}{4} for x_{1} in the equation x_{3}=\frac{3}{2}-\frac{5}{6}x_{1}.

$$x_{3}=\frac{3}{2}-\frac{5}{6}\left(-\frac{11}{8}x_{3}+\frac{3}{4}\right)$$

Solve x_{3}=\frac{3}{2}-\frac{5}{6}\left(-\frac{11}{8}x_{3}+\frac{3}{4}\right) for x_{3}.

$$x_{3}=-6$$

Substitute -6 for x_{3} in the equation x_{1}=-\frac{11}{8}x_{3}+\frac{3}{4}.

$$x_{1}=-\frac{11}{8}\left(-6\right)+\frac{3}{4}$$

Calculate x_{1} from x_{1}=-\frac{11}{8}\left(-6\right)+\frac{3}{4}.

$$x_{1}=9$$

Substitute 9 for x_{1} and -6 for x_{3} in the equation x_{2}=2x_{1}+3x_{3}-1.

$$x_{2}=2\times 9+3\left(-6\right)-1$$

Calculate x_{2} from x_{2}=2\times 9+3\left(-6\right)-1.

$$x_{2}=-1$$

The system is now solved.

$$x_{1}=9 x_{2}=-1 x_{3}=-6$$