Solve y/9-9/y/frac{9{y^2}-1/9}

$$\frac { \frac { y } { 9 } - \frac { 9 } { y } } { \frac { 9 } { y ^ { 2 } } - \frac { 1 } { 9 } }$$

Evaluate

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Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and y is 9y. Multiply \frac{y}{9} times \frac{y}{y}. Multiply \frac{9}{y} times \frac{9}{9}.

$$\frac{\frac{yy}{9y}-\frac{9\times 9}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

Since \frac{yy}{9y} and \frac{9\times 9}{9y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{yy-9\times 9}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

Do the multiplications in yy-9\times 9.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and 9 is 9y^{2}. Multiply \frac{9}{y^{2}} times \frac{9}{9}. Multiply \frac{1}{9} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9\times 9}{9y^{2}}-\frac{y^{2}}{9y^{2}}}$$

Since \frac{9\times 9}{9y^{2}} and \frac{y^{2}}{9y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9\times 9-y^{2}}{9y^{2}}}$$

Do the multiplications in 9\times 9-y^{2}.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{81-y^{2}}{9y^{2}}}$$

Divide \frac{y^{2}-81}{9y} by \frac{81-y^{2}}{9y^{2}} by multiplying \frac{y^{2}-81}{9y} by the reciprocal of \frac{81-y^{2}}{9y^{2}}.

$$\frac{\left(y^{2}-81\right)\times 9y^{2}}{9y\left(81-y^{2}\right)}$$

Extract the negative sign in y^{2}-81.

$$\frac{-9y^{2}\left(-y^{2}+81\right)}{9y\left(-y^{2}+81\right)}$$

Cancel out 9y\left(-y^{2}+81\right) in both numerator and denominator.

$$-y$$

Expand

$$-y$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and y is 9y. Multiply \frac{y}{9} times \frac{y}{y}. Multiply \frac{9}{y} times \frac{9}{9}.

$$\frac{\frac{yy}{9y}-\frac{9\times 9}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

Since \frac{yy}{9y} and \frac{9\times 9}{9y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{yy-9\times 9}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

Do the multiplications in yy-9\times 9.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9}{y^{2}}-\frac{1}{9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and 9 is 9y^{2}. Multiply \frac{9}{y^{2}} times \frac{9}{9}. Multiply \frac{1}{9} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9\times 9}{9y^{2}}-\frac{y^{2}}{9y^{2}}}$$

Since \frac{9\times 9}{9y^{2}} and \frac{y^{2}}{9y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{9\times 9-y^{2}}{9y^{2}}}$$

Do the multiplications in 9\times 9-y^{2}.

$$\frac{\frac{y^{2}-81}{9y}}{\frac{81-y^{2}}{9y^{2}}}$$

Divide \frac{y^{2}-81}{9y} by \frac{81-y^{2}}{9y^{2}} by multiplying \frac{y^{2}-81}{9y} by the reciprocal of \frac{81-y^{2}}{9y^{2}}.

$$\frac{\left(y^{2}-81\right)\times 9y^{2}}{9y\left(81-y^{2}\right)}$$

Extract the negative sign in y^{2}-81.

$$\frac{-9y^{2}\left(-y^{2}+81\right)}{9y\left(-y^{2}+81\right)}$$

Cancel out 9y\left(-y^{2}+81\right) in both numerator and denominator.

$$-y$$