Solve {l}{x+y+z=1}{2x+y-2z=1}{3x+y-z=11}

$$\left. \begin{array} { l } { x + y + z = 1 } \\ { 2 x + y - 2 z = 1 } \\ { 3 x + y - z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$y=-9$$

$$z = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Short Steps Using Substitution

Solve x+y+z=1 for x.

$$x=-y-z+1$$

Substitute -y-z+1 for x in the second and third equation.

$$2\left(-y-z+1\right)+y-2z=1 3\left(-y-z+1\right)+y-z=11$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=\frac{5}{2}$$

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

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

Calculate y from y=-4\times \frac{5}{2}+1.

$$y=-9$$

Substitute -9 for y and \frac{5}{2} for z in the equation x=-y-z+1.

$$x=-\left(-9\right)-\frac{5}{2}+1$$

Calculate x from x=-\left(-9\right)-\frac{5}{2}+1.

$$x=\frac{15}{2}$$

The system is now solved.

$$x=\frac{15}{2} y=-9 z=\frac{5}{2}$$