Solve {r}{2x_{1}+3x_{2}+5x_{3}-10=0}{3x_{1}+7x_{2}+4x_{3}-3=0}{x_{1}+2x_{2}+2x_{3}-3=0}

$$\left. \begin{array} { r } { 2 x _ { 1 } + 3 x _ { 2 } + 5 x _ { 3 } - 10 = 0 } \\ { 3 x _ { 1 } + 7 x _ { 2 } + 4 x _ { 3 } - 3 = 0 } \\ { x _ { 1 } + 2 x _ { 2 } + 2 x _ { 3 } - 3 = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=3$$

$$x_{2}=-2$$

$$x_{3}=2$$

Short Steps Using Substitution

Reorder the equations.

$$x_{1}+2x_{2}+2x_{3}-3=0 3x_{1}+7x_{2}+4x_{3}-3=0 2x_{1}+3x_{2}+5x_{3}-10=0$$

Solve x_{1}+2x_{2}+2x_{3}-3=0 for x_{1}.

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

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

$$3\left(-2x_{2}-2x_{3}+3\right)+7x_{2}+4x_{3}-3=0 2\left(-2x_{2}-2x_{3}+3\right)+3x_{2}+5x_{3}-10=0$$

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

$$x_{2}=-6+2x_{3} x_{3}=x_{2}+4$$

Substitute -6+2x_{3} for x_{2} in the equation x_{3}=x_{2}+4.

$$x_{3}=-6+2x_{3}+4$$

Solve x_{3}=-6+2x_{3}+4 for x_{3}.

$$x_{3}=2$$

Substitute 2 for x_{3} in the equation x_{2}=-6+2x_{3}.

$$x_{2}=-6+2\times 2$$

Calculate x_{2} from x_{2}=-6+2\times 2.

$$x_{2}=-2$$

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

$$x_{1}=-2\left(-2\right)-2\times 2+3$$

Calculate x_{1} from x_{1}=-2\left(-2\right)-2\times 2+3.

$$x_{1}=3$$

The system is now solved.

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