Solve {c}{2x_{1}+x_{2}+x_{3}=9}{x_{1}+x_{2}+x_{2}=5}{3x_{1}+x_{2}+x_{3}=17}

$$\left\{ \begin{array} { c } { 2 x _ { 1 } + x _ { 2 } + x _ { 3 } = 9 } \\ { x _ { 1 } + x _ { 2 } + x _ { 2 } = 5 } \\ { 3 x _ { 1 } + x _ { 2 } + x _ { 3 } = 17 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=8$$

$$x_{2} = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x_{3} = -\frac{11}{2} = -5\frac{1}{2} = -5.5$$

Short Steps Using Substitution

Solve 2x_{1}+x_{2}+x_{3}=9 for x_{2}.

$$x_{2}=-2x_{1}-x_{3}+9$$

Substitute -2x_{1}-x_{3}+9 for x_{2} in the second and third equation.

$$x_{1}-2x_{1}-x_{3}+9-2x_{1}-x_{3}+9=5 3x_{1}-2x_{1}-x_{3}+9+x_{3}=17$$

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

$$x_{3}=-\frac{3}{2}x_{1}+\frac{13}{2} x_{1}=8$$

Substitute 8 for x_{1} in the equation x_{3}=-\frac{3}{2}x_{1}+\frac{13}{2}.

$$x_{3}=-\frac{3}{2}\times 8+\frac{13}{2}$$

Calculate x_{3} from x_{3}=-\frac{3}{2}\times 8+\frac{13}{2}.

$$x_{3}=-\frac{11}{2}$$

Substitute -\frac{11}{2} for x_{3} and 8 for x_{1} in the equation x_{2}=-2x_{1}-x_{3}+9.

$$x_{2}=-2\times 8-\left(-\frac{11}{2}\right)+9$$

Calculate x_{2} from x_{2}=-2\times 8-\left(-\frac{11}{2}\right)+9.

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

The system is now solved.

$$x_{1}=8 x_{2}=-\frac{3}{2} x_{3}=-\frac{11}{2}$$