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

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

Solve for x, y, z

$$x=\frac{11w}{7}+1$$

$$y=\frac{45w}{14}+1$$

$$z=-\frac{13w}{14}$$

Short Steps Using Substitution

Solve 2x-y+z+w=1 for y.

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

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

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

Solve these equations for x and z respectively.

$$x=-6z-4w+1 z=\frac{3}{4}-\frac{3}{4}x+\frac{1}{4}w$$

Substitute -6z-4w+1 for x in the equation z=\frac{3}{4}-\frac{3}{4}x+\frac{1}{4}w.

$$z=\frac{3}{4}-\frac{3}{4}\left(-6z-4w+1\right)+\frac{1}{4}w$$

Solve z=\frac{3}{4}-\frac{3}{4}\left(-6z-4w+1\right)+\frac{1}{4}w for z.

$$z=-\frac{13}{14}w$$

Substitute -\frac{13}{14}w for z in the equation x=-6z-4w+1.

$$x=-6\left(-\frac{13}{14}\right)w-4w+1$$

Calculate x from x=-6\left(-\frac{13}{14}\right)w-4w+1.

$$x=1+\frac{11}{7}w$$

Substitute 1+\frac{11}{7}w for x and -\frac{13}{14}w for z in the equation y=2x+z+w-1.

$$y=2\left(1+\frac{11}{7}w\right)-\frac{13}{14}w+w-1$$

Calculate y from y=2\left(1+\frac{11}{7}w\right)-\frac{13}{14}w+w-1.

$$y=1+\frac{45}{14}w$$

The system is now solved.

$$x=1+\frac{11}{7}w y=1+\frac{45}{14}w z=-\frac{13}{14}w$$