Solve {l}{3x-2y=-1}{4x+z=-28}{x+2y+3z=-43}

$$\left\{ \begin{array} { l } { 3 x - 2 y = - 1 } \\ { 4 x + z = - 28 } \\ { x + 2 y + 3 z = - 43 } \end{array} \right.$$

Solve for x, y, z

$$x=-5$$

$$y=-7$$

$$z=-8$$

Short Steps Using Substitution

Reorder the equations.

$$4x+z=-28 3x-2y=-1 x+2y+3z=-43$$

Solve 4x+z=-28 for z.

$$z=-28-4x$$

Substitute -28-4x for z in the equation x+2y+3z=-43.

$$x+2y+3\left(-28-4x\right)=-43$$

Solve the second equation for y and the third equation for x.

$$y=\frac{3}{2}x+\frac{1}{2} x=\frac{2}{11}y-\frac{41}{11}$$

Substitute \frac{3}{2}x+\frac{1}{2} for y in the equation x=\frac{2}{11}y-\frac{41}{11}.

$$x=\frac{2}{11}\left(\frac{3}{2}x+\frac{1}{2}\right)-\frac{41}{11}$$

Solve x=\frac{2}{11}\left(\frac{3}{2}x+\frac{1}{2}\right)-\frac{41}{11} for x.

$$x=-5$$

Substitute -5 for x in the equation y=\frac{3}{2}x+\frac{1}{2}.

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

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

$$y=-7$$

Substitute -5 for x in the equation z=-28-4x.

$$z=-28-4\left(-5\right)$$

Calculate z from z=-28-4\left(-5\right).

$$z=-8$$

The system is now solved.

$$x=-5 y=-7 z=-8$$