Solve {c}{x+y+2z=6}{2y+z=-1}{x-y+4z=10}

$$\left\{ \begin{array} { c } { x + y + 2 z = 6 } \\ { 2 y + z = - 1 } \\ { x - y + 4 z = 10 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-1$$

$$z=1$$

Short Steps Using Substitution

Solve x+y+2z=6 for x.

$$x=-y-2z+6$$

Substitute -y-2z+6 for x in the equation x-y+4z=10.

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

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

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

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

$$z=2-\frac{1}{2}z-\frac{1}{2}$$

Solve z=2-\frac{1}{2}z-\frac{1}{2} for z.

$$z=1$$

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

$$y=-\frac{1}{2}-\frac{1}{2}$$

Calculate y from y=-\frac{1}{2}-\frac{1}{2}.

$$y=-1$$

Substitute -1 for y and 1 for z in the equation x=-y-2z+6.

$$x=-\left(-1\right)-2+6$$

Calculate x from x=-\left(-1\right)-2+6.

$$x=5$$

The system is now solved.

$$x=5 y=-1 z=1$$