Solve {l}{4x-2y-3z=8}{5x+3y-4z=4}{6x-4y-5z=12}

$$\left. \begin{array} { l } { 4 x - 2 y - 3 z = 8 } \\ { 5 x + 3 y - 4 z = 4 } \\ { 6 x - 4 y - 5 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

Solve 4x-2y-3z=8 for x.

$$x=\frac{1}{2}y+\frac{3}{4}z+2$$

Substitute \frac{1}{2}y+\frac{3}{4}z+2 for x in the second and third equation.

$$5\left(\frac{1}{2}y+\frac{3}{4}z+2\right)+3y-4z=4 6\left(\frac{1}{2}y+\frac{3}{4}z+2\right)-4y-5z=12$$

Solve these equations for y and z respectively.

$$y=-\frac{12}{11}+\frac{1}{22}z z=-2y$$

Substitute -\frac{12}{11}+\frac{1}{22}z for y in the equation z=-2y.

$$z=-2\left(-\frac{12}{11}+\frac{1}{22}z\right)$$

Solve z=-2\left(-\frac{12}{11}+\frac{1}{22}z\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=-\frac{12}{11}+\frac{1}{22}z.

$$y=-\frac{12}{11}+\frac{1}{22}\times 2$$

Calculate y from y=-\frac{12}{11}+\frac{1}{22}\times 2.

$$y=-1$$

Substitute -1 for y and 2 for z in the equation x=\frac{1}{2}y+\frac{3}{4}z+2.

$$x=\frac{1}{2}\left(-1\right)+\frac{3}{4}\times 2+2$$

Calculate x from x=\frac{1}{2}\left(-1\right)+\frac{3}{4}\times 2+2.

$$x=3$$

The system is now solved.

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