Solve {c}{-5x+6y-7z=25}{10x-11y+13z=-45}{-x+y-z=0}

$$\left\{ \begin{array} { c } { - 5 x + 6 y - 7 z = 25 } \\ { 10 x - 11 y + 13 z = - 45 } \\ { - x + y - z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-15$$

$$z=-20$$

Short Steps Using Substitution

Reorder the equations.

$$-x+y-z=0 10x-11y+13z=-45 -5x+6y-7z=25$$

Solve -x+y-z=0 for x.

$$x=y-z$$

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

$$10\left(y-z\right)-11y+13z=-45 -5\left(y-z\right)+6y-7z=25$$

Solve these equations for y and z respectively.

$$y=45+3z z=-\frac{25}{2}+\frac{1}{2}y$$

Substitute 45+3z for y in the equation z=-\frac{25}{2}+\frac{1}{2}y.

$$z=-\frac{25}{2}+\frac{1}{2}\left(45+3z\right)$$

Solve z=-\frac{25}{2}+\frac{1}{2}\left(45+3z\right) for z.

$$z=-20$$

Substitute -20 for z in the equation y=45+3z.

$$y=45+3\left(-20\right)$$

Calculate y from y=45+3\left(-20\right).

$$y=-15$$

Substitute -15 for y and -20 for z in the equation x=y-z.

$$x=-15-\left(-20\right)$$

Calculate x from x=-15-\left(-20\right).

$$x=5$$

The system is now solved.

$$x=5 y=-15 z=-20$$