Solve {l}{x-2y+3z=10}{2x+y-6z=1}{4x-2y-9z=15}

$$\left\{ \begin{array} { l } { x - 2 y + 3 z = 10 } \\ { 2 x + y - 6 z = 1 } \\ { 4 x - 2 y - 9 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-3$$

$$z=\frac{1}{3}\approx 0.333333333$$

Short Steps Using Substitution

Solve x-2y+3z=10 for x.

$$x=2y-3z+10$$

Substitute 2y-3z+10 for x in the second and third equation.

$$2\left(2y-3z+10\right)+y-6z=1 4\left(2y-3z+10\right)-2y-9z=15$$

Solve these equations for y and z respectively.

$$y=\frac{12}{5}z-\frac{19}{5} z=\frac{2}{7}y+\frac{25}{21}$$

Substitute \frac{12}{5}z-\frac{19}{5} for y in the equation z=\frac{2}{7}y+\frac{25}{21}.

$$z=\frac{2}{7}\left(\frac{12}{5}z-\frac{19}{5}\right)+\frac{25}{21}$$

Solve z=\frac{2}{7}\left(\frac{12}{5}z-\frac{19}{5}\right)+\frac{25}{21} for z.

$$z=\frac{1}{3}$$

Substitute \frac{1}{3} for z in the equation y=\frac{12}{5}z-\frac{19}{5}.

$$y=\frac{12}{5}\times \frac{1}{3}-\frac{19}{5}$$

Calculate y from y=\frac{12}{5}\times \frac{1}{3}-\frac{19}{5}.

$$y=-3$$

Substitute -3 for y and \frac{1}{3} for z in the equation x=2y-3z+10.

$$x=2\left(-3\right)-3\times \frac{1}{3}+10$$

Calculate x from x=2\left(-3\right)-3\times \frac{1}{3}+10.

$$x=3$$

The system is now solved.

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