Solve {l}{-5x-y+5z=1}{-4y+2z=-4}{7x+5y+5z=-53}

$$\left. \begin{array}{l}{ - 5 x - y + 5 z = 1 }\\{ - 4 y + 2 z = - 4 }\\{ 7 x + 5 y + 5 z = - 53 }\end{array} \right.$$

Solve for x, y, z

$$x=-4$$

$$y=-1$$

$$z=-4$$

Short Steps Using Substitution

Solve -5x-y+5z=1 for y.

$$y=-5x+5z-1$$

Substitute -5x+5z-1 for y in the second and third equation.

$$-4\left(-5x+5z-1\right)+2z=-4 7x+5\left(-5x+5z-1\right)+5z=-53$$

Solve these equations for x and z respectively.

$$x=-\frac{2}{5}+\frac{9}{10}z z=-\frac{8}{5}+\frac{3}{5}x$$

Substitute -\frac{2}{5}+\frac{9}{10}z for x in the equation z=-\frac{8}{5}+\frac{3}{5}x.

$$z=-\frac{8}{5}+\frac{3}{5}\left(-\frac{2}{5}+\frac{9}{10}z\right)$$

Solve z=-\frac{8}{5}+\frac{3}{5}\left(-\frac{2}{5}+\frac{9}{10}z\right) for z.

$$z=-4$$

Substitute -4 for z in the equation x=-\frac{2}{5}+\frac{9}{10}z.

$$x=-\frac{2}{5}+\frac{9}{10}\left(-4\right)$$

Calculate x from x=-\frac{2}{5}+\frac{9}{10}\left(-4\right).

$$x=-4$$

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

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

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

$$y=-1$$

The system is now solved.

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