Solve {l}{6x-2y+3w=-13}{5x+3y+2w=-2}{x-4y+6w=-4}

$$\left\{ \begin{array} { l } { 6 x - 2 y + 3 w = - 13 } \\ { 5 x + 3 y + 2 w = - 2 } \\ { x - 4 y + 6 w = - 4 } \end{array} \right.$$

Solve for x, y, w

$$x=-2$$

$$y=2$$

$$w=1$$

Short Steps Using Substitution

Reorder the equations.

$$x-4y+6w=-4 5x+3y+2w=-2 6x-2y+3w=-13$$

Solve x-4y+6w=-4 for x.

$$x=-4+4y-6w$$

Substitute -4+4y-6w for x in the second and third equation.

$$5\left(-4+4y-6w\right)+3y+2w=-2 6\left(-4+4y-6w\right)-2y+3w=-13$$

Solve these equations for y and w respectively.

$$y=\frac{18}{23}+\frac{28}{23}w w=-\frac{1}{3}+\frac{2}{3}y$$

Substitute \frac{18}{23}+\frac{28}{23}w for y in the equation w=-\frac{1}{3}+\frac{2}{3}y.

$$w=-\frac{1}{3}+\frac{2}{3}\left(\frac{18}{23}+\frac{28}{23}w\right)$$

Solve w=-\frac{1}{3}+\frac{2}{3}\left(\frac{18}{23}+\frac{28}{23}w\right) for w.

$$w=1$$

Substitute 1 for w in the equation y=\frac{18}{23}+\frac{28}{23}w.

$$y=\frac{18}{23}+\frac{28}{23}\times 1$$

Calculate y from y=\frac{18}{23}+\frac{28}{23}\times 1.

$$y=2$$

Substitute 2 for y and 1 for w in the equation x=-4+4y-6w.

$$x=-4+4\times 2-6$$

Calculate x from x=-4+4\times 2-6.

$$x=-2$$

The system is now solved.

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