Solve {l}{x-2y=5;2x+3z=12}{y+2z=3}

$$\left. \begin{array} { l } { x - 2 y = 5 ; 2 x + 3 z = 12 } \\ { y + 2 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

Solve x-2y=5 for x.

$$x=2y+5$$

Substitute 2y+5 for x in the equation 2x+3z=12.

$$2\left(2y+5\right)+3z=12$$

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

$$y=\frac{1}{2}-\frac{3}{4}z z=-\frac{1}{2}y+\frac{3}{2}$$

Substitute \frac{1}{2}-\frac{3}{4}z for y in the equation z=-\frac{1}{2}y+\frac{3}{2}.

$$z=-\frac{1}{2}\left(\frac{1}{2}-\frac{3}{4}z\right)+\frac{3}{2}$$

Solve z=-\frac{1}{2}\left(\frac{1}{2}-\frac{3}{4}z\right)+\frac{3}{2} for z.

$$z=2$$

Substitute 2 for z in the equation y=\frac{1}{2}-\frac{3}{4}z.

$$y=\frac{1}{2}-\frac{3}{4}\times 2$$

Calculate y from y=\frac{1}{2}-\frac{3}{4}\times 2.

$$y=-1$$

Substitute -1 for y in the equation x=2y+5.

$$x=2\left(-1\right)+5$$

Calculate x from x=2\left(-1\right)+5.

$$x=3$$

The system is now solved.

$$x=3 y=-1 z=2$$