Solve {l}{2x_{1}+3x_{2}+x_{3}=-1}{5x_{1}+x_{2}+x_{3}=9}{3x_{1}+2x_{2}+4x_{3}=11}

$$\left. \begin{array} { l } { 2 x _ { 1 } + 3 x _ { 2 } + x _ { 3 } = - 1 } \\ { 5 x _ { 1 } + x _ { 2 } + x _ { 3 } = 9 } \\ { 3 x _ { 1 } + 2 x _ { 2 } + 4 x _ { 3 } = 11 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

$$x_{2} = -\frac{19}{8} = -2\frac{3}{8} = -2.375$$

$$x_{3} = \frac{21}{8} = 2\frac{5}{8} = 2.625$$

Short Steps Using Substitution

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

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

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

$$5x_{1}+x_{2}-2x_{1}-3x_{2}-1=9 3x_{1}+2x_{2}+4\left(-2x_{1}-3x_{2}-1\right)=11$$

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

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

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

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

Solve x_{1}=-3-2\left(\frac{3}{2}x_{1}-5\right) for x_{1}.

$$x_{1}=\frac{7}{4}$$

Substitute \frac{7}{4} for x_{1} in the equation x_{2}=\frac{3}{2}x_{1}-5.

$$x_{2}=\frac{3}{2}\times \frac{7}{4}-5$$

Calculate x_{2} from x_{2}=\frac{3}{2}\times \frac{7}{4}-5.

$$x_{2}=-\frac{19}{8}$$

Substitute -\frac{19}{8} for x_{2} and \frac{7}{4} for x_{1} in the equation x_{3}=-2x_{1}-3x_{2}-1.

$$x_{3}=-2\times \frac{7}{4}-3\left(-\frac{19}{8}\right)-1$$

Calculate x_{3} from x_{3}=-2\times \frac{7}{4}-3\left(-\frac{19}{8}\right)-1.

$$x_{3}=\frac{21}{8}$$

The system is now solved.

$$x_{1}=\frac{7}{4} x_{2}=-\frac{19}{8} x_{3}=\frac{21}{8}$$