Solve (a+3b/a^2-3ab-1/a):b/3b-a

$$( \frac { a + 3 b } { a ^ { 2 } - 3 a b } - \frac { 1 } { a } ) : \frac { b } { 3 b - a }$$

Evaluate

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

Short Solution Steps

Factor a^{2}-3ab.

$$\frac{\frac{a+3b}{a\left(a-3b\right)}-\frac{1}{a}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b}{a\left(a-3b\right)}-\frac{a-3b}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b-\left(a-3b\right)}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b-a+3b}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

Combine like terms in a+3b-a+3b.

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

Divide \frac{6b}{a\left(a-3b\right)} by \frac{b}{3b-a} by multiplying \frac{6b}{a\left(a-3b\right)} by the reciprocal of \frac{b}{3b-a}.

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

Extract the negative sign in 3b-a.

$$\frac{-6b\left(a-3b\right)}{ab\left(a-3b\right)}$$

Cancel out b\left(a-3b\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor a^{2}-3ab.

$$\frac{\frac{a+3b}{a\left(a-3b\right)}-\frac{1}{a}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b}{a\left(a-3b\right)}-\frac{a-3b}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b-\left(a-3b\right)}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

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

$$\frac{\frac{a+3b-a+3b}{a\left(a-3b\right)}}{\frac{b}{3b-a}}$$

Combine like terms in a+3b-a+3b.

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

Divide \frac{6b}{a\left(a-3b\right)} by \frac{b}{3b-a} by multiplying \frac{6b}{a\left(a-3b\right)} by the reciprocal of \frac{b}{3b-a}.

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

Extract the negative sign in 3b-a.

$$\frac{-6b\left(a-3b\right)}{ab\left(a-3b\right)}$$

Cancel out b\left(a-3b\right) in both numerator and denominator.

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