Solve {l}{-3x+2y+4z=6}{6x-5y+6z=-3}{9x+6y-16z=-3}

$$\left. \begin{array} { l } { - 3 x + 2 y + 4 z = 6 } \\ { 6 x - 5 y + 6 z = - 3 } \\ { 9 x + 6 y - 16 z = - 3 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$z=\frac{3}{4}=0.75$$

Short Steps Using Substitution

Solve -3x+2y+4z=6 for x.

$$x=-2+\frac{2}{3}y+\frac{4}{3}z$$

Substitute -2+\frac{2}{3}y+\frac{4}{3}z for x in the second and third equation.

$$6\left(-2+\frac{2}{3}y+\frac{4}{3}z\right)-5y+6z=-3 9\left(-2+\frac{2}{3}y+\frac{4}{3}z\right)+6y-16z=-3$$

Solve these equations for y and z respectively.

$$y=-9+14z z=-\frac{15}{4}+3y$$

Substitute -9+14z for y in the equation z=-\frac{15}{4}+3y.

$$z=-\frac{15}{4}+3\left(-9+14z\right)$$

Solve z=-\frac{15}{4}+3\left(-9+14z\right) for z.

$$z=\frac{3}{4}$$

Substitute \frac{3}{4} for z in the equation y=-9+14z.

$$y=-9+14\times \frac{3}{4}$$

Calculate y from y=-9+14\times \frac{3}{4}.

$$y=\frac{3}{2}$$

Substitute \frac{3}{2} for y and \frac{3}{4} for z in the equation x=-2+\frac{2}{3}y+\frac{4}{3}z.

$$x=-2+\frac{2}{3}\times \frac{3}{2}+\frac{4}{3}\times \frac{3}{4}$$

Calculate x from x=-2+\frac{2}{3}\times \frac{3}{2}+\frac{4}{3}\times \frac{3}{4}.

$$x=0$$

The system is now solved.

$$x=0 y=\frac{3}{2} z=\frac{3}{4}$$