Solve {l}{-4x-y-2z=-37}{4x+6y+5z=100}{-6x-9y+z=-99}

$$\left. \begin{array} { l } { - 4 x - y - 2 z = - 37 } \\ { 4 x + 6 y + 5 z = 100 } \\ { - 6 x - 9 y + z = - 99 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=9$$

$$z=6$$

Short Steps Using Substitution

Solve -4x-y-2z=-37 for y.

$$y=-4x-2z+37$$

Substitute -4x-2z+37 for y in the second and third equation.

$$4x+6\left(-4x-2z+37\right)+5z=100 -6x-9\left(-4x-2z+37\right)+z=-99$$

Solve these equations for x and z respectively.

$$x=-\frac{7}{20}z+\frac{61}{10} z=\frac{234}{19}-\frac{30}{19}x$$

Substitute -\frac{7}{20}z+\frac{61}{10} for x in the equation z=\frac{234}{19}-\frac{30}{19}x.

$$z=\frac{234}{19}-\frac{30}{19}\left(-\frac{7}{20}z+\frac{61}{10}\right)$$

Solve z=\frac{234}{19}-\frac{30}{19}\left(-\frac{7}{20}z+\frac{61}{10}\right) for z.

$$z=6$$

Substitute 6 for z in the equation x=-\frac{7}{20}z+\frac{61}{10}.

$$x=-\frac{7}{20}\times 6+\frac{61}{10}$$

Calculate x from x=-\frac{7}{20}\times 6+\frac{61}{10}.

$$x=4$$

Substitute 4 for x and 6 for z in the equation y=-4x-2z+37.

$$y=-4\times 4-2\times 6+37$$

Calculate y from y=-4\times 4-2\times 6+37.

$$y=9$$

The system is now solved.

$$x=4 y=9 z=6$$