Solve 6a/a-5-3/6a-6

$$\frac { 6 a } { a - 5 } - \frac { 3 } { 6 a - 6 }$$

Evaluate

$$\frac{12a^{2}-13a+5}{2\left(a-5\right)\left(a-1\right)}$$

Short Solution Steps

Factor 6a-6.

$$\frac{6a}{a-5}-\frac{3}{6\left(a-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-5 and 6\left(a-1\right) is 6\left(a-5\right)\left(a-1\right). Multiply \frac{6a}{a-5} times \frac{6\left(a-1\right)}{6\left(a-1\right)}. Multiply \frac{3}{6\left(a-1\right)} times \frac{a-5}{a-5}.

$$\frac{6a\times 6\left(a-1\right)}{6\left(a-5\right)\left(a-1\right)}-\frac{3\left(a-5\right)}{6\left(a-5\right)\left(a-1\right)}$$

Since \frac{6a\times 6\left(a-1\right)}{6\left(a-5\right)\left(a-1\right)} and \frac{3\left(a-5\right)}{6\left(a-5\right)\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6a\times 6\left(a-1\right)-3\left(a-5\right)}{6\left(a-5\right)\left(a-1\right)}$$

Do the multiplications in 6a\times 6\left(a-1\right)-3\left(a-5\right).

$$\frac{36a^{2}-36a-3a+15}{6\left(a-5\right)\left(a-1\right)}$$

Combine like terms in 36a^{2}-36a-3a+15.

$$\frac{36a^{2}-39a+15}{6\left(a-5\right)\left(a-1\right)}$$

Factor the expressions that are not already factored in \frac{36a^{2}-39a+15}{6\left(a-5\right)\left(a-1\right)}.

$$\frac{3\left(12a^{2}-13a+5\right)}{6\left(a-5\right)\left(a-1\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{12a^{2}-13a+5}{2\left(a-5\right)\left(a-1\right)}$$

Expand 2\left(a-5\right)\left(a-1\right).

$$\frac{12a^{2}-13a+5}{2a^{2}-12a+10}$$

Factor

$$\frac{12a^{2}-13a+5}{2\left(a-5\right)\left(a-1\right)}$$