Solve {l}{4x+y+z=4}{2x-y+2z=8}{x+2y-z=-5}

$$\left\{ \begin{array} { l } { 4 x + y + z = 4 } \\ { 2 x - y + 2 z = 8 } \\ { x + 2 y - z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-2$$

$$z=2$$

Short Steps Using Substitution

Solve 4x+y+z=4 for y.

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

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

$$2x-\left(-4x-z+4\right)+2z=8 x+2\left(-4x-z+4\right)-z=-5$$

Solve these equations for x and z respectively.

$$x=2-\frac{1}{2}z z=-\frac{7}{3}x+\frac{13}{3}$$

Substitute 2-\frac{1}{2}z for x in the equation z=-\frac{7}{3}x+\frac{13}{3}.

$$z=-\frac{7}{3}\left(2-\frac{1}{2}z\right)+\frac{13}{3}$$

Solve z=-\frac{7}{3}\left(2-\frac{1}{2}z\right)+\frac{13}{3} for z.

$$z=2$$

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

$$x=2-\frac{1}{2}\times 2$$

Calculate x from x=2-\frac{1}{2}\times 2.

$$x=1$$

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

$$y=-4-2+4$$

Calculate y from y=-4-2+4.

$$y=-2$$

The system is now solved.

$$x=1 y=-2 z=2$$