Solve {l}{5x+2y-z=12}{x+3y-5z=-2}{7x+y-2z=12}

$$\left. \begin{array} { l } { 5 x + 2 y - z = 12 } \\ { x + 3 y - 5 z = - 2 } \\ { 7 x + y - 2 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=2$$

$$z=2$$

Short Steps Using Substitution

Solve 5x+2y-z=12 for z.

$$z=5x+2y-12$$

Substitute 5x+2y-12 for z in the second and third equation.

$$x+3y-5\left(5x+2y-12\right)=-2 7x+y-2\left(5x+2y-12\right)=12$$

Solve these equations for y and x respectively.

$$y=\frac{62}{7}-\frac{24}{7}x x=4-y$$

Substitute \frac{62}{7}-\frac{24}{7}x for y in the equation x=4-y.

$$x=4-\left(\frac{62}{7}-\frac{24}{7}x\right)$$

Solve x=4-\left(\frac{62}{7}-\frac{24}{7}x\right) for x.

$$x=2$$

Substitute 2 for x in the equation y=\frac{62}{7}-\frac{24}{7}x.

$$y=\frac{62}{7}-\frac{24}{7}\times 2$$

Calculate y from y=\frac{62}{7}-\frac{24}{7}\times 2.

$$y=2$$

Substitute 2 for y and 2 for x in the equation z=5x+2y-12.

$$z=5\times 2+2\times 2-12$$

Calculate z from z=5\times 2+2\times 2-12.

$$z=2$$

The system is now solved.

$$x=2 y=2 z=2$$