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

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

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=3$$

Short Steps Using Substitution

Solve 2x+4y+z=1 for z.

$$z=-2x-4y+1$$

Substitute -2x-4y+1 for z in the second and third equation.

$$x-2y-3\left(-2x-4y+1\right)=2 x+y-\left(-2x-4y+1\right)=-1$$

Solve these equations for y and x respectively.

$$y=-\frac{7}{10}x+\frac{1}{2} x=-\frac{5}{3}y$$

Substitute -\frac{7}{10}x+\frac{1}{2} for y in the equation x=-\frac{5}{3}y.

$$x=-\frac{5}{3}\left(-\frac{7}{10}x+\frac{1}{2}\right)$$

Solve x=-\frac{5}{3}\left(-\frac{7}{10}x+\frac{1}{2}\right) for x.

$$x=5$$

Substitute 5 for x in the equation y=-\frac{7}{10}x+\frac{1}{2}.

$$y=-\frac{7}{10}\times 5+\frac{1}{2}$$

Calculate y from y=-\frac{7}{10}\times 5+\frac{1}{2}.

$$y=-3$$

Substitute -3 for y and 5 for x in the equation z=-2x-4y+1.

$$z=-2\times 5-4\left(-3\right)+1$$

Calculate z from z=-2\times 5-4\left(-3\right)+1.

$$z=3$$

The system is now solved.

$$x=5 y=-3 z=3$$