Solve {l}{2x_{1}+2x_{2}+x_{3}=67000}{3x_{1}+x_{2}+x_{3}=61000}{x_{1}+3x_{2}+2x_{3}=80000}

$$\left. \begin{array} { l } { 2 x _ { 1 } + 2 x _ { 2 } + x _ { 3 } = 67000 } \\ { 3 x _ { 1 } + x _ { 2 } + x _ { 3 } = 61000 } \\ { x _ { 1 } + 3 x _ { 2 } + 2 x _ { 3 } = 80000 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=12000$$

$$x_{2}=18000$$

$$x_{3}=7000$$

Short Steps Using Substitution

Solve 2x_{1}+2x_{2}+x_{3}=67000 for x_{3}.

$$x_{3}=67000-2x_{1}-2x_{2}$$

Substitute 67000-2x_{1}-2x_{2} for x_{3} in the second and third equation.

$$3x_{1}+x_{2}+67000-2x_{1}-2x_{2}=61000 x_{1}+3x_{2}+2\left(67000-2x_{1}-2x_{2}\right)=80000$$

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

$$x_{2}=x_{1}+6000 x_{1}=-\frac{1}{3}x_{2}+18000$$

Substitute x_{1}+6000 for x_{2} in the equation x_{1}=-\frac{1}{3}x_{2}+18000.

$$x_{1}=-\frac{1}{3}\left(x_{1}+6000\right)+18000$$

Solve x_{1}=-\frac{1}{3}\left(x_{1}+6000\right)+18000 for x_{1}.

$$x_{1}=12000$$

Substitute 12000 for x_{1} in the equation x_{2}=x_{1}+6000.

$$x_{2}=12000+6000$$

Calculate x_{2} from x_{2}=12000+6000.

$$x_{2}=18000$$

Substitute 18000 for x_{2} and 12000 for x_{1} in the equation x_{3}=67000-2x_{1}-2x_{2}.

$$x_{3}=67000-2\times 12000-2\times 18000$$

Calculate x_{3} from x_{3}=67000-2\times 12000-2\times 18000.

$$x_{3}=7000$$

The system is now solved.

$$x_{1}=12000 x_{2}=18000 x_{3}=7000$$