Solve {l}{31x-10z=-15}{15y-6z=85}{6y=63}

$$\left\{ \begin{array} { l } { 31 x - 10 z = - 15 } \\ { 15 y - 6 z = 85 } \\ { 6 y = 63 } \end{array} \right.$$

Solve for x, z, y

$$x = \frac{635}{186} = 3\frac{77}{186} \approx 3.413978495$$

$$y = \frac{21}{2} = 10\frac{1}{2} = 10.5$$

$$z = \frac{145}{12} = 12\frac{1}{12} \approx 12.083333333$$

Solution Steps

Consider the third equation. Divide both sides by 6.

$$y=\frac{63}{6}$$

Reduce the fraction \frac{63}{6} to lowest terms by extracting and canceling out 3.

$$y=\frac{21}{2}$$

Consider the second equation. Insert the known values of variables into the equation.

$$15\times \frac{21}{2}-6z=85$$

Multiply 15 and \frac{21}{2} to get \frac{315}{2}.

$$\frac{315}{2}-6z=85$$

Subtract \frac{315}{2} from both sides.

$$-6z=85-\frac{315}{2}$$

Subtract \frac{315}{2} from 85 to get -\frac{145}{2}.

$$-6z=-\frac{145}{2}$$

Divide both sides by -6.

$$z=\frac{-\frac{145}{2}}{-6}$$

Express \frac{-\frac{145}{2}}{-6} as a single fraction.

$$z=\frac{-145}{2\left(-6\right)}$$

Multiply 2 and -6 to get -12.

$$z=\frac{-145}{-12}$$

Fraction \frac{-145}{-12} can be simplified to \frac{145}{12} by removing the negative sign from both the numerator and the denominator.

$$z=\frac{145}{12}$$

Consider the first equation. Insert the known values of variables into the equation.

$$31x-10\times \frac{145}{12}=-15$$

Multiply -10 and \frac{145}{12} to get -\frac{725}{6}.

$$31x-\frac{725}{6}=-15$$

Add \frac{725}{6} to both sides.

$$31x=-15+\frac{725}{6}$$

Add -15 and \frac{725}{6} to get \frac{635}{6}.

$$31x=\frac{635}{6}$$

Divide both sides by 31.

$$x=\frac{\frac{635}{6}}{31}$$

Express \frac{\frac{635}{6}}{31} as a single fraction.

$$x=\frac{635}{6\times 31}$$

Multiply 6 and 31 to get 186.

$$x=\frac{635}{186}$$

The system is now solved.

$$x=\frac{635}{186} z=\frac{145}{12} y=\frac{21}{2}$$