Solve {l}{3x+y+2z=25}{x+3y+z=12}{x+y+z=7}

$$\left\{ \begin{array} { l } { 3 x + y + 2 z = 25 } \\ { x + 3 y + z = 12 } \\ { x + y + z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{27}{2} = 13\frac{1}{2} = 13.5$$

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

$$z=-9$$

Short Steps Using Substitution

Solve 3x+y+2z=25 for y.

$$y=-3x-2z+25$$

Substitute -3x-2z+25 for y in the second and third equation.

$$x+3\left(-3x-2z+25\right)+z=12 x-3x-2z+25+z=7$$

Solve these equations for x and z respectively.

$$x=-\frac{5}{8}z+\frac{63}{8} z=-2x+18$$

Substitute -\frac{5}{8}z+\frac{63}{8} for x in the equation z=-2x+18.

$$z=-2\left(-\frac{5}{8}z+\frac{63}{8}\right)+18$$

Solve z=-2\left(-\frac{5}{8}z+\frac{63}{8}\right)+18 for z.

$$z=-9$$

Substitute -9 for z in the equation x=-\frac{5}{8}z+\frac{63}{8}.

$$x=-\frac{5}{8}\left(-9\right)+\frac{63}{8}$$

Calculate x from x=-\frac{5}{8}\left(-9\right)+\frac{63}{8}.

$$x=\frac{27}{2}$$

Substitute \frac{27}{2} for x and -9 for z in the equation y=-3x-2z+25.

$$y=-3\times \frac{27}{2}-2\left(-9\right)+25$$

Calculate y from y=-3\times \frac{27}{2}-2\left(-9\right)+25.

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

The system is now solved.

$$x=\frac{27}{2} y=\frac{5}{2} z=-9$$