Solve {l}{3x-y+2z=-5}{-2x+y-3z=4}{-4x+y-4z=9}

$$\left. \begin{array} { l } { 3 x - y + 2 z = - 5 } \\ { - 2 x + y - 3 z = 4 } \\ { - 4 x + y - 4 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=-3$$

$$z=-1$$

Short Steps Using Substitution

Solve 3x-y+2z=-5 for y.

$$y=3x+2z+5$$

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

$$-2x+3x+2z+5-3z=4 -4x+3x+2z+5-4z=9$$

Solve these equations for x and z respectively.

$$x=z-1 z=-\frac{1}{2}x-2$$

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

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

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

$$z=-1$$

Substitute -1 for z in the equation x=z-1.

$$x=-1-1$$

Calculate x from x=-1-1.

$$x=-2$$

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

$$y=3\left(-2\right)+2\left(-1\right)+5$$

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

$$y=-3$$

The system is now solved.

$$x=-2 y=-3 z=-1$$