Solve {l}{4x+2y-3z=8}{9x-3y+6z=12}{7x+5y-8z=-5}

$$\left. \begin{array} { l } { 4 x + 2 y - 3 z = 8 } \\ { 9 x - 3 y + 6 z = 12 } \\ { 7 x + 5 y - 8 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{17}{2} = -8\frac{1}{2} = -8.5$$

$$y = \frac{345}{2} = 172\frac{1}{2} = 172.5$$

$$z=101$$

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.

$$9\left(-\frac{1}{2}y+\frac{3}{4}z+2\right)-3y+6z=12 7\left(-\frac{1}{2}y+\frac{3}{4}z+2\right)+5y-8z=-5$$

Solve these equations for y and z respectively.

$$y=\frac{4}{5}+\frac{17}{10}z z=\frac{76}{11}+\frac{6}{11}y$$

Substitute \frac{4}{5}+\frac{17}{10}z for y in the equation z=\frac{76}{11}+\frac{6}{11}y.

$$z=\frac{76}{11}+\frac{6}{11}\left(\frac{4}{5}+\frac{17}{10}z\right)$$

Solve z=\frac{76}{11}+\frac{6}{11}\left(\frac{4}{5}+\frac{17}{10}z\right) for z.

$$z=101$$

Substitute 101 for z in the equation y=\frac{4}{5}+\frac{17}{10}z.

$$y=\frac{4}{5}+\frac{17}{10}\times 101$$

Calculate y from y=\frac{4}{5}+\frac{17}{10}\times 101.

$$y=\frac{345}{2}$$

Substitute \frac{345}{2} for y and 101 for z in the equation x=-\frac{1}{2}y+\frac{3}{4}z+2.

$$x=-\frac{1}{2}\times \frac{345}{2}+\frac{3}{4}\times 101+2$$

Calculate x from x=-\frac{1}{2}\times \frac{345}{2}+\frac{3}{4}\times 101+2.

$$x=-\frac{17}{2}$$

The system is now solved.

$$x=-\frac{17}{2} y=\frac{345}{2} z=101$$