Solve {l}{x_{1}+x_{2}+x_{3}=10000}{12x_{1}+15x_{2}+22x_{3}=200000}{2x_{1}=x_{2}}

$$\left\{ \begin{array} { l } { x _ { 1 } + x _ { 2 } + x _ { 3 } = 10000 } \\ { 12 x _ { 1 } + 15 x _ { 2 } + 22 x _ { 3 } = 200000 } \\ { 2 x _ { 1 } = x _ { 2 } } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = \frac{2500}{3} = 833\frac{1}{3} \approx 833.333333333$$

$$x_{2} = \frac{5000}{3} = 1666\frac{2}{3} \approx 1666.666666667$$

$$x_{3}=7500$$

Short Steps Using Substitution

Reorder the equations.

$$2x_{1}=x_{2} 12x_{1}+15x_{2}+22x_{3}=200000 x_{1}+x_{2}+x_{3}=10000$$

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

$$x_{2}=2x_{1}$$

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

$$12x_{1}+15\times 2x_{1}+22x_{3}=200000 x_{1}+2x_{1}+x_{3}=10000$$

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

$$x_{1}=\frac{100000}{21}-\frac{11}{21}x_{3} x_{3}=-3x_{1}+10000$$

Substitute \frac{100000}{21}-\frac{11}{21}x_{3} for x_{1} in the equation x_{3}=-3x_{1}+10000.

$$x_{3}=-3\left(\frac{100000}{21}-\frac{11}{21}x_{3}\right)+10000$$

Solve x_{3}=-3\left(\frac{100000}{21}-\frac{11}{21}x_{3}\right)+10000 for x_{3}.

$$x_{3}=7500$$

Substitute 7500 for x_{3} in the equation x_{1}=\frac{100000}{21}-\frac{11}{21}x_{3}.

$$x_{1}=\frac{100000}{21}-\frac{11}{21}\times 7500$$

Calculate x_{1} from x_{1}=\frac{100000}{21}-\frac{11}{21}\times 7500.

$$x_{1}=\frac{2500}{3}$$

Substitute \frac{2500}{3} for x_{1} in the equation x_{2}=2x_{1}.

$$x_{2}=2\times \frac{2500}{3}$$

Calculate x_{2} from x_{2}=2\times \frac{2500}{3}.

$$x_{2}=\frac{5000}{3}$$

The system is now solved.

$$x_{1}=\frac{2500}{3} x_{2}=\frac{5000}{3} x_{3}=7500$$