Solve {r}{2x_{1}+3x_{2}-x_{3}=5}{-4x_{1}+6x_{2}+8x_{3}=7}{16x_{1}+12x_{2}+14x_{3}=9}

$$\left. \begin{array} { r } { 2 x _ { 1 } + 3 x _ { 2 } - x _ { 3 } = 5 } \\ { - 4 x _ { 1 } + 6 x _ { 2 } + 8 x _ { 3 } = 7 } \\ { 16 x _ { 1 } + 12 x _ { 2 } + 14 x _ { 3 } = 9 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=-\frac{1}{4}=-0.25$$

$$x_{2} = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$x_{3}=-\frac{1}{2}=-0.5$$

Short Steps Using Substitution

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

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

Substitute 2x_{1}+3x_{2}-5 for x_{3} in the second and third equation.

$$-4x_{1}+6x_{2}+8\left(2x_{1}+3x_{2}-5\right)=7 16x_{1}+12x_{2}+14\left(2x_{1}+3x_{2}-5\right)=9$$

Solve these equations for x_{2} and x_{1} respectively.

$$x_{2}=-\frac{2}{5}x_{1}+\frac{47}{30} x_{1}=-\frac{27}{22}x_{2}+\frac{79}{44}$$

Substitute -\frac{2}{5}x_{1}+\frac{47}{30} for x_{2} in the equation x_{1}=-\frac{27}{22}x_{2}+\frac{79}{44}.

$$x_{1}=-\frac{27}{22}\left(-\frac{2}{5}x_{1}+\frac{47}{30}\right)+\frac{79}{44}$$

Solve x_{1}=-\frac{27}{22}\left(-\frac{2}{5}x_{1}+\frac{47}{30}\right)+\frac{79}{44} for x_{1}.

$$x_{1}=-\frac{1}{4}$$

Substitute -\frac{1}{4} for x_{1} in the equation x_{2}=-\frac{2}{5}x_{1}+\frac{47}{30}.

$$x_{2}=-\frac{2}{5}\left(-\frac{1}{4}\right)+\frac{47}{30}$$

Calculate x_{2} from x_{2}=-\frac{2}{5}\left(-\frac{1}{4}\right)+\frac{47}{30}.

$$x_{2}=\frac{5}{3}$$

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

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

Calculate x_{3} from x_{3}=2\left(-\frac{1}{4}\right)+3\times \frac{5}{3}-5.

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

The system is now solved.

$$x_{1}=-\frac{1}{4} x_{2}=\frac{5}{3} x_{3}=-\frac{1}{2}$$