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

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

Solve for x, y, z

$$x = \frac{110}{27} = 4\frac{2}{27} \approx 4.074074074$$

$$y = \frac{589}{108} = 5\frac{49}{108} \approx 5.453703704$$

$$z = \frac{335}{108} = 3\frac{11}{108} \approx 3.101851852$$

Short Steps Using Substitution

Solve 5x-9y+7z=-7 for x.

$$x=\frac{9}{5}y-\frac{7}{5}z-\frac{7}{5}$$

Substitute \frac{9}{5}y-\frac{7}{5}z-\frac{7}{5} for x in the equation 7x-6y+2z=2.

$$7\left(\frac{9}{5}y-\frac{7}{5}z-\frac{7}{5}\right)-6y+2z=2$$

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

$$y=\frac{59}{33}+\frac{13}{11}z z=\frac{1}{2}y+\frac{3}{8}$$

Substitute \frac{59}{33}+\frac{13}{11}z for y in the equation z=\frac{1}{2}y+\frac{3}{8}.

$$z=\frac{1}{2}\left(\frac{59}{33}+\frac{13}{11}z\right)+\frac{3}{8}$$

Solve z=\frac{1}{2}\left(\frac{59}{33}+\frac{13}{11}z\right)+\frac{3}{8} for z.

$$z=\frac{335}{108}$$

Substitute \frac{335}{108} for z in the equation y=\frac{59}{33}+\frac{13}{11}z.

$$y=\frac{59}{33}+\frac{13}{11}\times \frac{335}{108}$$

Calculate y from y=\frac{59}{33}+\frac{13}{11}\times \frac{335}{108}.

$$y=\frac{589}{108}$$

Substitute \frac{589}{108} for y and \frac{335}{108} for z in the equation x=\frac{9}{5}y-\frac{7}{5}z-\frac{7}{5}.

$$x=\frac{9}{5}\times \frac{589}{108}-\frac{7}{5}\times \frac{335}{108}-\frac{7}{5}$$

Calculate x from x=\frac{9}{5}\times \frac{589}{108}-\frac{7}{5}\times \frac{335}{108}-\frac{7}{5}.

$$x=\frac{110}{27}$$

The system is now solved.

$$x=\frac{110}{27} y=\frac{589}{108} z=\frac{335}{108}$$