Solve a+7/a-7+a+9/a-6

$$\frac{ a+7 }{ a-7 } + \frac{ a+9 }{ a-6 }$$

Evaluate

$$\frac{2a^{2}+3a-105}{\left(a-7\right)\left(a-6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-7 and a-6 is \left(a-7\right)\left(a-6\right). Multiply \frac{a+7}{a-7} times \frac{a-6}{a-6}. Multiply \frac{a+9}{a-6} times \frac{a-7}{a-7}.

$$\frac{\left(a+7\right)\left(a-6\right)}{\left(a-7\right)\left(a-6\right)}+\frac{\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)}$$

Since \frac{\left(a+7\right)\left(a-6\right)}{\left(a-7\right)\left(a-6\right)} and \frac{\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+7\right)\left(a-6\right)+\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)}$$

Do the multiplications in \left(a+7\right)\left(a-6\right)+\left(a+9\right)\left(a-7\right).

$$\frac{a^{2}-6a+7a-42+a^{2}-7a+9a-63}{\left(a-7\right)\left(a-6\right)}$$

Combine like terms in a^{2}-6a+7a-42+a^{2}-7a+9a-63.

$$\frac{2a^{2}+3a-105}{\left(a-7\right)\left(a-6\right)}$$

Expand \left(a-7\right)\left(a-6\right).

$$\frac{2a^{2}+3a-105}{a^{2}-13a+42}$$

Expand

$$\frac{2a^{2}+3a-105}{\left(a-7\right)\left(a-6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-7 and a-6 is \left(a-7\right)\left(a-6\right). Multiply \frac{a+7}{a-7} times \frac{a-6}{a-6}. Multiply \frac{a+9}{a-6} times \frac{a-7}{a-7}.

$$\frac{\left(a+7\right)\left(a-6\right)}{\left(a-7\right)\left(a-6\right)}+\frac{\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)}$$

Since \frac{\left(a+7\right)\left(a-6\right)}{\left(a-7\right)\left(a-6\right)} and \frac{\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+7\right)\left(a-6\right)+\left(a+9\right)\left(a-7\right)}{\left(a-7\right)\left(a-6\right)}$$

Do the multiplications in \left(a+7\right)\left(a-6\right)+\left(a+9\right)\left(a-7\right).

$$\frac{a^{2}-6a+7a-42+a^{2}-7a+9a-63}{\left(a-7\right)\left(a-6\right)}$$

Combine like terms in a^{2}-6a+7a-42+a^{2}-7a+9a-63.

$$\frac{2a^{2}+3a-105}{\left(a-7\right)\left(a-6\right)}$$

Expand \left(a-7\right)\left(a-6\right).

$$\frac{2a^{2}+3a-105}{a^{2}-13a+42}$$