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

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

Solve for x, y, z

$$x=3$$

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

$$z=-\frac{1}{2}=-0.5$$

Short Steps Using Substitution

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

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

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

$$x+2x+z-4+z=4 3x-2\left(2x+z-4\right)+2z=5$$

Solve these equations for z and x respectively.

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

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

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

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

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

Substitute -\frac{1}{2} for z and 3 for x in the equation y=2x+z-4.

$$y=2\times 3-\frac{1}{2}-4$$

Calculate y from y=2\times 3-\frac{1}{2}-4.

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

The system is now solved.

$$x=3 y=\frac{3}{2} z=-\frac{1}{2}$$