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

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

Solve for x_1, x_2, x_3

$$x_{1}=-4$$

$$x_{2}=\frac{1}{7}\approx 0.142857143$$

$$x_{3} = \frac{16}{7} = 2\frac{2}{7} \approx 2.285714286$$

Short Steps Using Substitution

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

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

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

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

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

$$x_{3}=2+2x_{2} x_{2}=\frac{1}{7}$$

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

$$x_{3}=2+2\times \frac{1}{7}$$

Calculate x_{3} from x_{3}=2+2\times \frac{1}{7}.

$$x_{3}=\frac{16}{7}$$

Substitute \frac{16}{7} for x_{3} and \frac{1}{7} for x_{2} in the equation x_{1}=-3x_{2}-2x_{3}+1.

$$x_{1}=-3\times \frac{1}{7}-2\times \frac{16}{7}+1$$

Calculate x_{1} from x_{1}=-3\times \frac{1}{7}-2\times \frac{16}{7}+1.

$$x_{1}=-4$$

The system is now solved.

$$x_{1}=-4 x_{2}=\frac{1}{7} x_{3}=\frac{16}{7}$$