Solve {l}{4y-2z=10}{2x+2y+2z=-6}{x+4y-2z=9}

$$\left. \begin{array} { l } { 4 y - 2 z = 10 } \\ { 2 x + 2 y + 2 z = - 6 } \\ { x + 4 y - 2 z = 9 } \end{array} \right.$$

Solve for y, z, x

$$x=-1$$

$$y=1$$

$$z=-3$$

Short Steps Using Substitution

Reorder the equations.

$$x+4y-2z=9 2x+2y+2z=-6 4y-2z=10$$

Solve x+4y-2z=9 for x.

$$x=-4y+2z+9$$

Substitute -4y+2z+9 for x in the equation 2x+2y+2z=-6.

$$2\left(-4y+2z+9\right)+2y+2z=-6$$

Solve the second equation for z and the third equation for y.

$$z=-4+y y=\frac{5}{2}+\frac{1}{2}z$$

Substitute -4+y for z in the equation y=\frac{5}{2}+\frac{1}{2}z.

$$y=\frac{5}{2}+\frac{1}{2}\left(-4+y\right)$$

Solve y=\frac{5}{2}+\frac{1}{2}\left(-4+y\right) for y.

$$y=1$$

Substitute 1 for y in the equation z=-4+y.

$$z=-4+1$$

Calculate z from z=-4+1.

$$z=-3$$

Substitute -3 for z and 1 for y in the equation x=-4y+2z+9.

$$x=-4+2\left(-3\right)+9$$

Calculate x from x=-4+2\left(-3\right)+9.

$$x=-1$$

The system is now solved.

$$y=1 z=-3 x=-1$$