Solve {l}{-6x+5y+2z=-18}{5x-6y-6z=5}{-2x-3y=10}

$$\left\{ \begin{array} { l } { - 6 x + 5 y + 2 z = - 18 } \\ { 5 x - 6 y - 6 z = 5 } \\ { - 2 x - 3 y = 10 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-4$$

$$z=4$$

Short Steps Using Substitution

Solve -6x+5y+2z=-18 for x.

$$x=\frac{5}{6}y+\frac{1}{3}z+3$$

Substitute \frac{5}{6}y+\frac{1}{3}z+3 for x in the second and third equation.

$$5\left(\frac{5}{6}y+\frac{1}{3}z+3\right)-6y-6z=5 -2\left(\frac{5}{6}y+\frac{1}{3}z+3\right)-3y=10$$

Solve these equations for y and z respectively.

$$y=\frac{60}{11}-\frac{26}{11}z z=-24-7y$$

Substitute \frac{60}{11}-\frac{26}{11}z for y in the equation z=-24-7y.

$$z=-24-7\left(\frac{60}{11}-\frac{26}{11}z\right)$$

Solve z=-24-7\left(\frac{60}{11}-\frac{26}{11}z\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=\frac{60}{11}-\frac{26}{11}z.

$$y=\frac{60}{11}-\frac{26}{11}\times 4$$

Calculate y from y=\frac{60}{11}-\frac{26}{11}\times 4.

$$y=-4$$

Substitute -4 for y and 4 for z in the equation x=\frac{5}{6}y+\frac{1}{3}z+3.

$$x=\frac{5}{6}\left(-4\right)+\frac{1}{3}\times 4+3$$

Calculate x from x=\frac{5}{6}\left(-4\right)+\frac{1}{3}\times 4+3.

$$x=1$$

The system is now solved.

$$x=1 y=-4 z=4$$