Math Problem Solver

The system of equations has \(2\) solutions.

\[x = - \frac{\sqrt{30}}{10}, y = - \frac{\sqrt{30}}{10}, z = - \frac{\sqrt{30}}{20}\]\[x = \frac{\sqrt{30}}{10}, y = \frac{\sqrt{30}}{10}, z = \frac{\sqrt{30}}{20}\]

Solve \(0 = 2 x + \frac{y}{2} - \frac{3}{8 z}\) for \(x\). \[x = - \frac{y}{4} + \frac{3}{16 z}\]Substitute \(- \frac{y}{4} + \frac{3}{16 z}\) for \(x\) in each of the remaining equations and simplify. $$\begin{aligned}0 &amp= \frac{x}{2} + 2 y - \frac{3}{8 z} \\ 0 &= \frac{\left(- \frac{y}{4} + \frac{3}{16 z}\right)}{2} + 2 y - \frac{3}{8 z} \\ \frac{3 \cdot \left(20 y z - 3\right)}{32 z} &= 0 \end{aligned}$$$$\begin{aligned}0 &amp= - \frac{3}{8 z} + \frac{3}{8 y} + \frac{3}{8 x} \\ 0 &= - \frac{3}{8 z} + \frac{3}{8 y} + \frac{3}{8 \left(- \frac{y}{4} + \frac{3}{16 z}\right)} \\ \frac{3 \left(- 4 y^{2} z - 12 z^{2} y + 3 y - 3 z\right)}{8 y z \left(4 y z - 3\right)} &= 0 \end{aligned}$$Solve \(\frac{3 \cdot \left(20 y z - 3\right)}{32 z} = 0\) for \(y\). \[y = \frac{3}{20 z}\]Substitute \(\frac{3}{20 z}\) for \(y\) in \(\frac{3 \left(- 4 y^{2} z - 12 z^{2} y + 3 y - 3 z\right)}{8 y z \left(4 y z - 3\right)} = 0\) and simplify. $$\begin{aligned}\frac{3 \left(- 4 y^{2} z - 12 z^{2} y + 3 y - 3 z\right)}{8 y z \left(4 y z - 3\right)} &amp= 0 \\ \frac{3 \left(- 4 \left(\frac{3}{20 z}\right)^{2} z - 12 z^{2} \left(\frac{3}{20 z}\right) + 3 \left(\frac{3}{20 z}\right) - 3 z\right)}{8 \left(\frac{3}{20 z}\right) z \left(4 \left(\frac{3}{20 z}\right) z - 3\right)} &= 0 \\ 5 z - \frac{3}{8 z} &= 0 \end{aligned}$$Substitute \(- \frac{\sqrt{30}}{20}\) into \(\frac{3 \cdot \left(20 y z - 3\right)}{32 z} = 0\) to solve for \(y\). \[\begin{aligned}\frac{15 y}{8} + \frac{3 \sqrt{30}}{16} &= 0\\\frac{15 y}{8} &= - \frac{3 \sqrt{30}}{16}\\y &= - \frac{\sqrt{30}}{10}\end{aligned}\]Substitute \(- \frac{\sqrt{30}}{20}\) for \(z\) and \(- \frac{\sqrt{30}}{10}\) for \(y\) in \(x = - \frac{y}{4} + \frac{3}{16 z}\) to solve for \(x\).$$\begin{aligned} x &= - \frac{y}{4} + \frac{3}{16 z}\\x &= \frac{3}{16 \left(- \frac{\sqrt{30}}{20}\right)} - \frac{\left(-1\right) \frac{1}{10} \sqrt{30}}{4}\\x&=- \frac{\sqrt{30}}{10}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = - \frac{\sqrt{30}}{10},\,y = - \frac{\sqrt{30}}{10},\,z = - \frac{\sqrt{30}}{20}\end{aligned}$$Substitute \(\frac{\sqrt{30}}{20}\) into \(\frac{3 \cdot \left(20 y z - 3\right)}{32 z} = 0\) to solve for \(y\). \[\begin{aligned}\frac{15 y}{8} - \frac{3 \sqrt{30}}{16} &= 0\\\frac{15 y}{8} &= \frac{3 \sqrt{30}}{16}\\y &= \frac{\sqrt{30}}{10}\end{aligned}\]Substitute \(\frac{\sqrt{30}}{20}\) for \(z\) and \(\frac{\sqrt{30}}{10}\) for \(y\) in \(x = - \frac{y}{4} + \frac{3}{16 z}\) to solve for \(x\).$$\begin{aligned} x &= - \frac{y}{4} + \frac{3}{16 z}\\x &= - \frac{\frac{1}{10} \sqrt{30}}{4} + \frac{3}{16 \frac{\sqrt{30}}{20}}\\x&=\frac{\sqrt{30}}{10}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{\sqrt{30}}{10},\,y = \frac{\sqrt{30}}{10},\,z = \frac{\sqrt{30}}{20}\end{aligned}$$