Solve {l}{-x_{1}+4x_{2}+2x_{3}=3}{3x_{1}+x_{1}+x_{3}=5}{6x_{1}+7x_{3}=12}

$$\left\{ \begin{array} { l } { - x _ { 1 } + 4 x _ { 2 } + 2 x _ { 3 } = 3 } \\ { 3 x _ { 1 } + x _ { 1 } + x _ { 3 } = 5 } \\ { 6 x _ { 1 } + 7 x _ { 3 } = 12 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{23}{22} = 1\frac{1}{22} \approx 1.045454545$$

$$x_{2}=\frac{53}{88}\approx 0.602272727$$

$$x_{3}=\frac{9}{11}\approx 0.818181818$$

Short Steps Using Substitution

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

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

Substitute 4x_{2}+2x_{3}-3 for x_{1} in the second and third equation.

$$3\left(4x_{2}+2x_{3}-3\right)+4x_{2}+2x_{3}-3+x_{3}=5 6\left(4x_{2}+2x_{3}-3\right)+7x_{3}=12$$

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

$$x_{2}=\frac{17}{16}-\frac{9}{16}x_{3} x_{3}=\frac{30}{19}-\frac{24}{19}x_{2}$$

Substitute \frac{17}{16}-\frac{9}{16}x_{3} for x_{2} in the equation x_{3}=\frac{30}{19}-\frac{24}{19}x_{2}.

$$x_{3}=\frac{30}{19}-\frac{24}{19}\left(\frac{17}{16}-\frac{9}{16}x_{3}\right)$$

Solve x_{3}=\frac{30}{19}-\frac{24}{19}\left(\frac{17}{16}-\frac{9}{16}x_{3}\right) for x_{3}.

$$x_{3}=\frac{9}{11}$$

Substitute \frac{9}{11} for x_{3} in the equation x_{2}=\frac{17}{16}-\frac{9}{16}x_{3}.

$$x_{2}=\frac{17}{16}-\frac{9}{16}\times \frac{9}{11}$$

Calculate x_{2} from x_{2}=\frac{17}{16}-\frac{9}{16}\times \frac{9}{11}.

$$x_{2}=\frac{53}{88}$$

Substitute \frac{53}{88} for x_{2} and \frac{9}{11} for x_{3} in the equation x_{1}=4x_{2}+2x_{3}-3.

$$x_{1}=4\times \frac{53}{88}+2\times \frac{9}{11}-3$$

Calculate x_{1} from x_{1}=4\times \frac{53}{88}+2\times \frac{9}{11}-3.

$$x_{1}=\frac{23}{22}$$

The system is now solved.

$$x_{1}=\frac{23}{22} x_{2}=\frac{53}{88} x_{3}=\frac{9}{11}$$