Solve {l}{x+y-z=5}{x-3y+z=7}{3x+6y-8z=6}

$$\left. \begin{array} { l } { x + y - z = 5 } \\ { x - 3 y + z = 7 } \\ { 3 x + 6 y - 8 z = 6 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{46}{7} = 6\frac{4}{7} \approx 6.571428571$$

$$y=\frac{4}{7}\approx 0.571428571$$

$$z = \frac{15}{7} = 2\frac{1}{7} \approx 2.142857143$$

Short Steps Using Substitution

Solve x+y-z=5 for x.

$$x=-y+z+5$$

Substitute -y+z+5 for x in the second and third equation.

$$-y+z+5-3y+z=7 3\left(-y+z+5\right)+6y-8z=6$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{2}+\frac{1}{2}z z=\frac{9}{5}+\frac{3}{5}y$$

Substitute -\frac{1}{2}+\frac{1}{2}z for y in the equation z=\frac{9}{5}+\frac{3}{5}y.

$$z=\frac{9}{5}+\frac{3}{5}\left(-\frac{1}{2}+\frac{1}{2}z\right)$$

Solve z=\frac{9}{5}+\frac{3}{5}\left(-\frac{1}{2}+\frac{1}{2}z\right) for z.

$$z=\frac{15}{7}$$

Substitute \frac{15}{7} for z in the equation y=-\frac{1}{2}+\frac{1}{2}z.

$$y=-\frac{1}{2}+\frac{1}{2}\times \frac{15}{7}$$

Calculate y from y=-\frac{1}{2}+\frac{1}{2}\times \frac{15}{7}.

$$y=\frac{4}{7}$$

Substitute \frac{4}{7} for y and \frac{15}{7} for z in the equation x=-y+z+5.

$$x=-\frac{4}{7}+\frac{15}{7}+5$$

Calculate x from x=-\frac{4}{7}+\frac{15}{7}+5.

$$x=\frac{46}{7}$$

The system is now solved.

$$x=\frac{46}{7} y=\frac{4}{7} z=\frac{15}{7}$$