Solve {l}{x_{1}-2x_{2}+x_{3}=1}{4x_{1}-3x_{2}+x_{3}=3}{2x_{1}-5x_{2}-3x_{3}=-9}

$$\left\{ \begin{array} { l } { x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = 1 } \\ { 4 x _ { 1 } - 3 x _ { 2 } + x _ { 3 } = 3 } \\ { 2 x _ { 1 } - 5 x _ { 2 } - 3 x _ { 3 } = - 9 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=1$$

$$x_{2}=1$$

$$x_{3}=2$$

Short Steps Using Substitution

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

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

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

$$4\left(2x_{2}-x_{3}+1\right)-3x_{2}+x_{3}=3 2\left(2x_{2}-x_{3}+1\right)-5x_{2}-3x_{3}=-9$$

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

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

Substitute \frac{3}{5}x_{3}-\frac{1}{5} for x_{2} in the equation x_{3}=-\frac{1}{5}x_{2}+\frac{11}{5}.

$$x_{3}=-\frac{1}{5}\left(\frac{3}{5}x_{3}-\frac{1}{5}\right)+\frac{11}{5}$$

Solve x_{3}=-\frac{1}{5}\left(\frac{3}{5}x_{3}-\frac{1}{5}\right)+\frac{11}{5} for x_{3}.

$$x_{3}=2$$

Substitute 2 for x_{3} in the equation x_{2}=\frac{3}{5}x_{3}-\frac{1}{5}.

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

Calculate x_{2} from x_{2}=\frac{3}{5}\times 2-\frac{1}{5}.

$$x_{2}=1$$

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

$$x_{1}=2\times 1-2+1$$

Calculate x_{1} from x_{1}=2\times 1-2+1.

$$x_{1}=1$$

The system is now solved.

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