Solve {l}{3x+y+5z+10w=2}{x+y+2z+4w=0}{x-2y+4z+w=3}

$$\left. \begin{array} { l } { 3 x + y + 5 z + 10 w = 2 } \\ { x + y + 2 z + 4 w = 0 } \\ { x - 2 y + 4 z + w = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=1-3w$$

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

$$z=0$$

Short Steps Using Substitution

Solve 3x+y+5z+10w=2 for y.

$$y=-3x-5z-10w+2$$

Substitute -3x-5z-10w+2 for y in the second and third equation.

$$x-3x-5z-10w+2+2z+4w=0 x-2\left(-3x-5z-10w+2\right)+4z+w=3$$

Solve these equations for x and z respectively.

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

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

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

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

$$z=0$$

Substitute 0 for z in the equation x=-\frac{3}{2}z-3w+1.

$$x=-\frac{3}{2}\times 0-3w+1$$

Calculate x from x=-\frac{3}{2}\times 0-3w+1.

$$x=1-3w$$

Substitute 1-3w for x and 0 for z in the equation y=-3x-5z-10w+2.

$$y=-3\left(1-3w\right)-5\times 0-10w+2$$

Calculate y from y=-3\left(1-3w\right)-5\times 0-10w+2.

$$y=-1-w$$

The system is now solved.

$$x=1-3w y=-1-w z=0$$