Solve {l}{y-2z=1}{-x+3y=3}{-x+5z=2}

$$\left\{ \begin{array} { l } { y - 2 z = 1 } \\ { - x + 3 y = 3 } \\ { - x + 5 z = 2 } \end{array} \right.$$

Solve for y, z, x

$$x=-12$$

$$y=-3$$

$$z=-2$$

Short Steps Using Substitution

Solve y-2z=1 for y.

$$y=2z+1$$

Substitute 2z+1 for y in the equation -x+3y=3.

$$-x+3\left(2z+1\right)=3$$

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

$$z=\frac{1}{6}x x=5z-2$$

Substitute \frac{1}{6}x for z in the equation x=5z-2.

$$x=5\times \frac{1}{6}x-2$$

Solve x=5\times \frac{1}{6}x-2 for x.

$$x=-12$$

Substitute -12 for x in the equation z=\frac{1}{6}x.

$$z=\frac{1}{6}\left(-12\right)$$

Calculate z from z=\frac{1}{6}\left(-12\right).

$$z=-2$$

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

$$y=2\left(-2\right)+1$$

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

$$y=-3$$

The system is now solved.

$$y=-3 z=-2 x=-12$$