Solve {c}{2x+y-3z=-9}{x-2y+4z=17}{3x-y-7=2}

$$\left. \begin{array} { c } { 2 x + y - 3 z = - 9 } \\ { x - 2 y + 4 z = 17 } \\ { 3 x - y - 7 = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{3}{5}=-0.6$$

$$y = -\frac{54}{5} = -10\frac{4}{5} = -10.8$$

$$z=-1$$

Short Steps Using Substitution

Solve 2x+y-3z=-9 for y.

$$y=-2x+3z-9$$

Substitute -2x+3z-9 for y in the second and third equation.

$$x-2\left(-2x+3z-9\right)+4z=17 3x-\left(-2x+3z-9\right)-7=2$$

Solve these equations for x and z respectively.

$$x=\frac{2}{5}z-\frac{1}{5} z=\frac{5}{3}x$$

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

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

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

$$z=-1$$

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

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

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

$$x=-\frac{3}{5}$$

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

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

Calculate y from y=-2\left(-\frac{3}{5}\right)+3\left(-1\right)-9.

$$y=-\frac{54}{5}$$

The system is now solved.

$$x=-\frac{3}{5} y=-\frac{54}{5} z=-1$$