Solve {l}{4x-y+5z=-16}{-3y+4z=-14}{-7x+4y+6z=3}

$$\left. \begin{array}{l}{ 4 x - y + 5 z = - 16 }\\{ - 3 y + 4 z = - 14 }\\{ - 7 x + 4 y + 6 z = 3 }\end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=2$$

$$z=-2$$

Short Steps Using Substitution

Solve 4x-y+5z=-16 for y.

$$y=4x+5z+16$$

Substitute 4x+5z+16 for y in the second and third equation.

$$-3\left(4x+5z+16\right)+4z=-14 -7x+4\left(4x+5z+16\right)+6z=3$$

Solve these equations for x and z respectively.

$$x=-\frac{17}{6}-\frac{11}{12}z z=-\frac{9}{26}x-\frac{61}{26}$$

Substitute -\frac{17}{6}-\frac{11}{12}z for x in the equation z=-\frac{9}{26}x-\frac{61}{26}.

$$z=-\frac{9}{26}\left(-\frac{17}{6}-\frac{11}{12}z\right)-\frac{61}{26}$$

Solve z=-\frac{9}{26}\left(-\frac{17}{6}-\frac{11}{12}z\right)-\frac{61}{26} for z.

$$z=-2$$

Substitute -2 for z in the equation x=-\frac{17}{6}-\frac{11}{12}z.

$$x=-\frac{17}{6}-\frac{11}{12}\left(-2\right)$$

Calculate x from x=-\frac{17}{6}-\frac{11}{12}\left(-2\right).

$$x=-1$$

Substitute -1 for x and -2 for z in the equation y=4x+5z+16.

$$y=4\left(-1\right)+5\left(-2\right)+16$$

Calculate y from y=4\left(-1\right)+5\left(-2\right)+16.

$$y=2$$

The system is now solved.

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