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

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

Solve for x, y, z

$$x = \frac{58}{19} = 3\frac{1}{19} \approx 3.052631579$$

$$y = -\frac{50}{19} = -2\frac{12}{19} \approx -2.631578947$$

$$z = \frac{66}{19} = 3\frac{9}{19} \approx 3.473684211$$

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.

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

Solve these equations for y and x respectively.

$$y=34-12x x=\frac{24}{7}+\frac{1}{7}y$$

Substitute 34-12x for y in the equation x=\frac{24}{7}+\frac{1}{7}y.

$$x=\frac{24}{7}+\frac{1}{7}\left(34-12x\right)$$

Solve x=\frac{24}{7}+\frac{1}{7}\left(34-12x\right) for x.

$$x=\frac{58}{19}$$

Substitute \frac{58}{19} for x in the equation y=34-12x.

$$y=34-12\times \frac{58}{19}$$

Calculate y from y=34-12\times \frac{58}{19}.

$$y=-\frac{50}{19}$$

Substitute -\frac{50}{19} for y and \frac{58}{19} for x in the equation z=-5x+2y+24.

$$z=-5\times \frac{58}{19}+2\left(-\frac{50}{19}\right)+24$$

Calculate z from z=-5\times \frac{58}{19}+2\left(-\frac{50}{19}\right)+24.

$$z=\frac{66}{19}$$

The system is now solved.

$$x=\frac{58}{19} y=-\frac{50}{19} z=\frac{66}{19}$$