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

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

Solve for x, y, z

$$x = \frac{69}{19} = 3\frac{12}{19} \approx 3.631578947$$

$$y = -\frac{182}{19} = -9\frac{11}{19} \approx -9.578947368$$

$$z = -\frac{253}{19} = -13\frac{6}{19} \approx -13.315789474$$

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.

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

Solve these equations for y and x respectively.

$$y=34-12x x=5+\frac{1}{7}y$$

Substitute 34-12x for y in the equation x=5+\frac{1}{7}y.

$$x=5+\frac{1}{7}\left(34-12x\right)$$

Solve x=5+\frac{1}{7}\left(34-12x\right) for x.

$$x=\frac{69}{19}$$

Substitute \frac{69}{19} for x in the equation y=34-12x.

$$y=34-12\times \frac{69}{19}$$

Calculate y from y=34-12\times \frac{69}{19}.

$$y=-\frac{182}{19}$$

Substitute -\frac{182}{19} for y and \frac{69}{19} for x in the equation z=-5x+2y+24.

$$z=-5\times \frac{69}{19}+2\left(-\frac{182}{19}\right)+24$$

Calculate z from z=-5\times \frac{69}{19}+2\left(-\frac{182}{19}\right)+24.

$$z=-\frac{253}{19}$$

The system is now solved.

$$x=\frac{69}{19} y=-\frac{182}{19} z=-\frac{253}{19}$$