Solve {l}{-3x+y=-10}{3x+2y+3z=4}{4x-y-2z=15}

$$\left\{ \begin{array} { l } { - 3 x + y = - 10 } \\ { 3 x + 2 y + 3 z = 4 } \\ { 4 x - y - 2 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-1$$

$$z=-1$$

Short Steps Using Substitution

Solve -3x+y=-10 for y.

$$y=3x-10$$

Substitute 3x-10 for y in the second and third equation.

$$3x+2\left(3x-10\right)+3z=4 4x-\left(3x-10\right)-2z=15$$

Solve these equations for x and z respectively.

$$x=\frac{8}{3}-\frac{1}{3}z z=-\frac{5}{2}+\frac{1}{2}x$$

Substitute \frac{8}{3}-\frac{1}{3}z for x in the equation z=-\frac{5}{2}+\frac{1}{2}x.

$$z=-\frac{5}{2}+\frac{1}{2}\left(\frac{8}{3}-\frac{1}{3}z\right)$$

Solve z=-\frac{5}{2}+\frac{1}{2}\left(\frac{8}{3}-\frac{1}{3}z\right) for z.

$$z=-1$$

Substitute -1 for z in the equation x=\frac{8}{3}-\frac{1}{3}z.

$$x=\frac{8}{3}-\frac{1}{3}\left(-1\right)$$

Calculate x from x=\frac{8}{3}-\frac{1}{3}\left(-1\right).

$$x=3$$

Substitute 3 for x in the equation y=3x-10.

$$y=3\times 3-10$$

Calculate y from y=3\times 3-10.

$$y=-1$$

The system is now solved.

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