Solve {l}{3/4x^{2}-1/2y^{2}-1/3z^2}{+1/3y^2+frac{1}{4}z^2-frac{1}{6}x^2}

$$\left. \begin{array} { l } { \frac { 3 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 3 } z ^ { 2 } } \\ { + \frac { 1 } { 3 } y ^ { 2 } + \frac { 1 } { 4 } z ^ { 2 } - \frac { 1 } { 6 } x ^ { 2 } } \end{array} \right.$$

Evaluate

$$\frac{7x^{2}}{12}-\frac{y^{2}}{6}-\frac{z^{2}}{12}$$

Solution Steps

Combine -\frac{1}{2}y^{2} and \frac{1}{3}y^{2} to get -\frac{1}{6}y^{2}.

$$\frac{3}{4}x^{2}-\frac{1}{6}y^{2}-\frac{1}{3}z^{2}+\frac{1}{4}z^{2}-\frac{1}{6}x^{2}$$

Combine -\frac{1}{3}z^{2} and \frac{1}{4}z^{2} to get -\frac{1}{12}z^{2}.

$$\frac{3}{4}x^{2}-\frac{1}{6}y^{2}-\frac{1}{12}z^{2}-\frac{1}{6}x^{2}$$

Combine \frac{3}{4}x^{2} and -\frac{1}{6}x^{2} to get \frac{7}{12}x^{2}.

$$\frac{7}{12}x^{2}-\frac{1}{6}y^{2}-\frac{1}{12}z^{2}$$

Factor

$$\frac{7x^{2}-2y^{2}-z^{2}}{12}$$

Solution Steps

Factor out \frac{1}{12}.

$$\frac{9x^{2}-6y^{2}-4z^{2}+4y^{2}+3z^{2}-2x^{2}}{12}$$