Solve {l}{5x-2y+z=24}{2x+5y-2z=-14}{x-4y+3z=26}

$$\left\{ \begin{array} { l } { 5 x - 2 y + z = 24 } \\ { 2 x + 5 y - 2 z = - 14 } \\ { x - 4 y + 3 z = 26 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-2$$

$$z=5$$

Short Steps Using Substitution

Solve 5x-2y+z=24 for z.

$$z=-5x+2y+24$$

Substitute -5x+2y+24 for z in the second and third equation.

$$2x+5y-2\left(-5x+2y+24\right)=-14 x-4y+3\left(-5x+2y+24\right)=26$$

Solve these equations for y and x respectively.

$$y=34-12x x=\frac{23}{7}+\frac{1}{7}y$$

Substitute 34-12x for y in the equation x=\frac{23}{7}+\frac{1}{7}y.

$$x=\frac{23}{7}+\frac{1}{7}\left(34-12x\right)$$

Solve x=\frac{23}{7}+\frac{1}{7}\left(34-12x\right) for x.

$$x=3$$

Substitute 3 for x in the equation y=34-12x.

$$y=34-12\times 3$$

Calculate y from y=34-12\times 3.

$$y=-2$$

Substitute -2 for y and 3 for x in the equation z=-5x+2y+24.

$$z=-5\times 3+2\left(-2\right)+24$$

Calculate z from z=-5\times 3+2\left(-2\right)+24.

$$z=5$$

The system is now solved.

$$x=3 y=-2 z=5$$