Solve {l}{2x-5y+4z=-3}{x-2y+z=5}{x-4y+6z=10}

$$\left. \begin{array}{l}{ 2 x - 5 y + 4 z = - 3 }\\{ x - 2 y + z = 5 }\\{ x - 4 y + 6 z = 10 }\end{array} \right.$$

Solve for x, y, z

$$x=124$$

$$y=75$$

$$z=31$$

Short Steps Using Substitution

Reorder the equations.

$$x-2y+z=5 2x-5y+4z=-3 x-4y+6z=10$$

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

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

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

$$2\left(2y-z+5\right)-5y+4z=-3 2y-z+5-4y+6z=10$$

Solve these equations for y and z respectively.

$$y=2z+13 z=\frac{2}{5}y+1$$

Substitute 2z+13 for y in the equation z=\frac{2}{5}y+1.

$$z=\frac{2}{5}\left(2z+13\right)+1$$

Solve z=\frac{2}{5}\left(2z+13\right)+1 for z.

$$z=31$$

Substitute 31 for z in the equation y=2z+13.

$$y=2\times 31+13$$

Calculate y from y=2\times 31+13.

$$y=75$$

Substitute 75 for y and 31 for z in the equation x=2y-z+5.

$$x=2\times 75-31+5$$

Calculate x from x=2\times 75-31+5.

$$x=124$$

The system is now solved.

$$x=124 y=75 z=31$$