Solve (a-y/a-b-b-y/a+b):1/a^2-b^2

$$( \frac { a - y } { a - b } - \frac { b - y } { a + b } ) : \frac { 1 } { a ^ { 2 } - b ^ { 2 } }$$

Evaluate

$$b^{2}+a^{2}-2by$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and a+b is \left(a+b\right)\left(a-b\right). Multiply \frac{a-y}{a-b} times \frac{a+b}{a+b}. Multiply \frac{b-y}{a+b} times \frac{a-b}{a-b}.

$$\frac{\frac{\left(a-y\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Since \frac{\left(a-y\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a-y\right)\left(a+b\right)-\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Do the multiplications in \left(a-y\right)\left(a+b\right)-\left(b-y\right)\left(a-b\right).

$$\frac{\frac{a^{2}+ab-ya-yb-ba+b^{2}+ya-yb}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Combine like terms in a^{2}+ab-ya-yb-ba+b^{2}+ya-yb.

$$\frac{\frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Divide \frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)} by \frac{1}{a^{2}-b^{2}} by multiplying \frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{1}{a^{2}-b^{2}}.

$$\frac{\left(a^{2}+b^{2}-2yb\right)\left(a^{2}-b^{2}\right)}{\left(a+b\right)\left(a-b\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a+b\right)\left(a-b\right)\left(-2by+a^{2}+b^{2}\right)}{\left(a+b\right)\left(a-b\right)}$$

Cancel out \left(a+b\right)\left(a-b\right) in both numerator and denominator.

$$-2by+a^{2}+b^{2}$$

Expand

$$b^{2}+a^{2}-2by$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and a+b is \left(a+b\right)\left(a-b\right). Multiply \frac{a-y}{a-b} times \frac{a+b}{a+b}. Multiply \frac{b-y}{a+b} times \frac{a-b}{a-b}.

$$\frac{\frac{\left(a-y\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Since \frac{\left(a-y\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a-y\right)\left(a+b\right)-\left(b-y\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Do the multiplications in \left(a-y\right)\left(a+b\right)-\left(b-y\right)\left(a-b\right).

$$\frac{\frac{a^{2}+ab-ya-yb-ba+b^{2}+ya-yb}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Combine like terms in a^{2}+ab-ya-yb-ba+b^{2}+ya-yb.

$$\frac{\frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)}}{\frac{1}{a^{2}-b^{2}}}$$

Divide \frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)} by \frac{1}{a^{2}-b^{2}} by multiplying \frac{a^{2}+b^{2}-2yb}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{1}{a^{2}-b^{2}}.

$$\frac{\left(a^{2}+b^{2}-2yb\right)\left(a^{2}-b^{2}\right)}{\left(a+b\right)\left(a-b\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a+b\right)\left(a-b\right)\left(-2by+a^{2}+b^{2}\right)}{\left(a+b\right)\left(a-b\right)}$$

Cancel out \left(a+b\right)\left(a-b\right) in both numerator and denominator.

$$-2by+a^{2}+b^{2}$$