Solve {l}{2x+y+z=12}{2x-y+z=7}{x+2y-z=6}

$$\left. \begin{array} { l } { 2 x + y + z = 12 } \\ { 2 x - y + z = 7 } \\ { x + 2 y - z = 6 } \end{array} \right.$$

Solve for x, y, z

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

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

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

Short Steps Using Substitution

Solve 2x+y+z=12 for y.

$$y=-2x-z+12$$

Substitute -2x-z+12 for y in the second and third equation.

$$2x-\left(-2x-z+12\right)+z=7 x+2\left(-2x-z+12\right)-z=6$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{2}z+\frac{19}{4} z=6-x$$

Substitute -\frac{1}{2}z+\frac{19}{4} for x in the equation z=6-x.

$$z=6-\left(-\frac{1}{2}z+\frac{19}{4}\right)$$

Solve z=6-\left(-\frac{1}{2}z+\frac{19}{4}\right) for z.

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

Substitute \frac{5}{2} for z in the equation x=-\frac{1}{2}z+\frac{19}{4}.

$$x=-\frac{1}{2}\times \frac{5}{2}+\frac{19}{4}$$

Calculate x from x=-\frac{1}{2}\times \frac{5}{2}+\frac{19}{4}.

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

Substitute \frac{7}{2} for x and \frac{5}{2} for z in the equation y=-2x-z+12.

$$y=-2\times \frac{7}{2}-\frac{5}{2}+12$$

Calculate y from y=-2\times \frac{7}{2}-\frac{5}{2}+12.

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

The system is now solved.

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