Solve 3a-6b/a+b-7a-6b/a-b-4a-5b/a+b-7a-8b/a-b

$$\frac { 3 a - 6 b } { a + b } - \frac { 7 a - 6 b } { a - b } - \frac { 4 a - 5 b } { a + b } - \frac { 7 a - 8 b } { a - b }$$

Evaluate

$$-15$$

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{3a-6b}{a+b} times \frac{a-b}{a-b}. Multiply \frac{7a-6b}{a-b} times \frac{a+b}{a+b}.

$$\frac{\left(3a-6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{\left(7a-6b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{4a-5b}{a+b}-\frac{7a-8b}{a-b}$$

Since \frac{\left(3a-6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{\left(7a-6b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3a-6b\right)\left(a-b\right)-\left(7a-6b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{4a-5b}{a+b}-\frac{7a-8b}{a-b}$$

Do the multiplications in \left(3a-6b\right)\left(a-b\right)-\left(7a-6b\right)\left(a+b\right).

$$\frac{3a^{2}-3ab-6ba+6b^{2}-7a^{2}-7ab+6ba+6b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{4a-5b}{a+b}-\frac{7a-8b}{a-b}$$

Combine like terms in 3a^{2}-3ab-6ba+6b^{2}-7a^{2}-7ab+6ba+6b^{2}.

$$\frac{-4a^{2}+12b^{2}-10ab}{\left(a+b\right)\left(a-b\right)}-\frac{4a-5b}{a+b}-\frac{7a-8b}{a-b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(a-b\right) and a+b is \left(a+b\right)\left(a-b\right). Multiply \frac{4a-5b}{a+b} times \frac{a-b}{a-b}.

$$\frac{-4a^{2}+12b^{2}-10ab}{\left(a+b\right)\left(a-b\right)}-\frac{\left(4a-5b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Since \frac{-4a^{2}+12b^{2}-10ab}{\left(a+b\right)\left(a-b\right)} and \frac{\left(4a-5b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4a^{2}+12b^{2}-10ab-\left(4a-5b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Do the multiplications in -4a^{2}+12b^{2}-10ab-\left(4a-5b\right)\left(a-b\right).

$$\frac{-4a^{2}+12b^{2}-10ab-4a^{2}+4ab+5ba-5b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Combine like terms in -4a^{2}+12b^{2}-10ab-4a^{2}+4ab+5ba-5b^{2}.

$$\frac{-8a^{2}-ab+7b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Factor the expressions that are not already factored in \frac{-8a^{2}-ab+7b^{2}}{\left(a+b\right)\left(a-b\right)}.

$$\frac{\left(8a-7b\right)\left(-a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Extract the negative sign in -a-b.

$$\frac{-\left(a+b\right)\left(8a-7b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{7a-8b}{a-b}$$

Cancel out a+b in both numerator and denominator.

$$\frac{-\left(8a-7b\right)}{a-b}-\frac{7a-8b}{a-b}$$

Since \frac{-\left(8a-7b\right)}{a-b} and \frac{7a-8b}{a-b} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(8a-7b\right)-\left(7a-8b\right)}{a-b}$$

Do the multiplications in -\left(8a-7b\right)-\left(7a-8b\right).

$$\frac{-8a+7b-7a+8b}{a-b}$$

Combine like terms in -8a+7b-7a+8b.

$$\frac{-15a+15b}{a-b}$$

Factor the expressions that are not already factored in \frac{-15a+15b}{a-b}.

$$\frac{15\left(-a+b\right)}{a-b}$$

Extract the negative sign in -a+b.

$$\frac{-15\left(a-b\right)}{a-b}$$

Cancel out a-b in both numerator and denominator.

$$-15$$

Factor

$$-15$$