Solve {l}{x+y+z=3}{2x+y-z=1}{x+2y+2z=1}{text{inMatrixform.}}

$$\left. \begin{array} { l } { x + y + z = 3 } \\ { 2 x + y - z = 1 } \\ { x + 2 y + 2 z = 1 } \\ { \text { in Matrix form. } } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y = -\frac{11}{2} = -5\frac{1}{2} = -5.5$$

$$z = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Short Steps Using Substitution

Solve x+y+z=3 for x.

$$x=-y-z+3$$

Substitute -y-z+3 for x in the second and third equation.

$$2\left(-y-z+3\right)+y-z=1 -y-z+3+2y+2z=1$$

Solve these equations for y and z respectively.

$$y=-3z+5 z=-y-2$$

Substitute -3z+5 for y in the equation z=-y-2.

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

Solve z=-\left(-3z+5\right)-2 for z.

$$z=\frac{7}{2}$$

Substitute \frac{7}{2} for z in the equation y=-3z+5.

$$y=-3\times \frac{7}{2}+5$$

Calculate y from y=-3\times \frac{7}{2}+5.

$$y=-\frac{11}{2}$$

Substitute -\frac{11}{2} for y and \frac{7}{2} for z in the equation x=-y-z+3.

$$x=-\left(-\frac{11}{2}\right)-\frac{7}{2}+3$$

Calculate x from x=-\left(-\frac{11}{2}\right)-\frac{7}{2}+3.

$$x=5$$

The system is now solved.

$$x=5 y=-\frac{11}{2} z=\frac{7}{2}$$