Solve {l}{3x+y=-1}{2y-4z=0}{x+2z=1}

$$\left. \begin{array} { l } { 3 x + y = - 1 } \\ { 2 y - 4 z = 0 } \\ { x + 2 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve 3x+y=-1 for y.

$$y=-3x-1$$

Substitute -3x-1 for y in the equation 2y-4z=0.

$$2\left(-3x-1\right)-4z=0$$

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

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

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

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

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

$$z=1$$

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

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

Calculate x from x=-\frac{1}{3}-\frac{2}{3}.

$$x=-1$$

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

$$y=-3\left(-1\right)-1$$

Calculate y from y=-3\left(-1\right)-1.

$$y=2$$

The system is now solved.

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