Solve {c}{9x+8y+2z=10}{3x+11y-9z=1}{3+10x+8y=11}

$$\left\{ \begin{array} { c } { 9 x + 8 y + 2 z = 10 } \\ { 3 x + 11 y - 9 z = 1 } \\ { 3 + 10 x + 8 y = 11 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{4}{61}\approx 0.06557377$$

$$y=\frac{56}{61}\approx 0.918032787$$

$$z = \frac{63}{61} = 1\frac{2}{61} \approx 1.032786885$$

Short Steps Using Substitution

Solve 9x+8y+2z=10 for x.

$$x=\frac{10}{9}-\frac{8}{9}y-\frac{2}{9}z$$

Substitute \frac{10}{9}-\frac{8}{9}y-\frac{2}{9}z for x in the second and third equation.

$$3\left(\frac{10}{9}-\frac{8}{9}y-\frac{2}{9}z\right)+11y-9z=1 3+10\left(\frac{10}{9}-\frac{8}{9}y-\frac{2}{9}z\right)+8y=11$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{25}+\frac{29}{25}z z=\frac{7}{5}-\frac{2}{5}y$$

Substitute -\frac{7}{25}+\frac{29}{25}z for y in the equation z=\frac{7}{5}-\frac{2}{5}y.

$$z=\frac{7}{5}-\frac{2}{5}\left(-\frac{7}{25}+\frac{29}{25}z\right)$$

Solve z=\frac{7}{5}-\frac{2}{5}\left(-\frac{7}{25}+\frac{29}{25}z\right) for z.

$$z=\frac{63}{61}$$

Substitute \frac{63}{61} for z in the equation y=-\frac{7}{25}+\frac{29}{25}z.

$$y=-\frac{7}{25}+\frac{29}{25}\times \frac{63}{61}$$

Calculate y from y=-\frac{7}{25}+\frac{29}{25}\times \frac{63}{61}.

$$y=\frac{56}{61}$$

Substitute \frac{56}{61} for y and \frac{63}{61} for z in the equation x=\frac{10}{9}-\frac{8}{9}y-\frac{2}{9}z.

$$x=\frac{10}{9}-\frac{8}{9}\times \frac{56}{61}-\frac{2}{9}\times \frac{63}{61}$$

Calculate x from x=\frac{10}{9}-\frac{8}{9}\times \frac{56}{61}-\frac{2}{9}\times \frac{63}{61}.

$$x=\frac{4}{61}$$

The system is now solved.

$$x=\frac{4}{61} y=\frac{56}{61} z=\frac{63}{61}$$