Solve {l}{4x-9z=17}{3x+y+15z=18}{x+2y+3z=2}

$$\left. \begin{array} { l } { 4 x - 9 z = 17 } \\ { 3 x + y + 15 z = 18 } \\ { x + 2 y + 3 z = 2 } \end{array} \right.$$

Solve for x, z, y

$$x=5$$

$$y=-2$$

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

Short Steps Using Substitution

Reorder the equations.

$$3x+y+15z=18 4x-9z=17 x+2y+3z=2$$

Solve 3x+y+15z=18 for y.

$$y=18-3x-15z$$

Substitute 18-3x-15z for y in the equation x+2y+3z=2.

$$x+2\left(18-3x-15z\right)+3z=2$$

Solve the second equation for z and the third equation for x.

$$z=\frac{4}{9}x-\frac{17}{9} x=\frac{34}{5}-\frac{27}{5}z$$

Substitute \frac{4}{9}x-\frac{17}{9} for z in the equation x=\frac{34}{5}-\frac{27}{5}z.

$$x=\frac{34}{5}-\frac{27}{5}\left(\frac{4}{9}x-\frac{17}{9}\right)$$

Solve x=\frac{34}{5}-\frac{27}{5}\left(\frac{4}{9}x-\frac{17}{9}\right) for x.

$$x=5$$

Substitute 5 for x in the equation z=\frac{4}{9}x-\frac{17}{9}.

$$z=\frac{4}{9}\times 5-\frac{17}{9}$$

Calculate z from z=\frac{4}{9}\times 5-\frac{17}{9}.

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

Substitute \frac{1}{3} for z and 5 for x in the equation y=18-3x-15z.

$$y=18-3\times 5-15\times \frac{1}{3}$$

Calculate y from y=18-3\times 5-15\times \frac{1}{3}.

$$y=-2$$

The system is now solved.

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