Solve 1/6x-4y-r/6x+4y+3x/4y^2-9x^2

$$\frac { 1 } { 6 x - 4 y } - \frac { r } { 6 x + 4 y } + \frac { 3 x } { 4 y ^ { 2 } - 9 x ^ { 2 } }$$

Evaluate

$$-\frac{r+1}{2\left(3x+2y\right)}$$

Short Solution Steps

Factor 6x-4y. Factor 6x+4y.

$$\frac{1}{2\left(3x-2y\right)}-\frac{r}{2\left(3x+2y\right)}+\frac{3x}{4y^{2}-9x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(3x-2y\right) and 2\left(3x+2y\right) is 2\left(3x-2y\right)\left(3x+2y\right). Multiply \frac{1}{2\left(3x-2y\right)} times \frac{3x+2y}{3x+2y}. Multiply \frac{r}{2\left(3x+2y\right)} times \frac{3x-2y}{3x-2y}.

$$\frac{3x+2y}{2\left(3x-2y\right)\left(3x+2y\right)}-\frac{r\left(3x-2y\right)}{2\left(3x-2y\right)\left(3x+2y\right)}+\frac{3x}{4y^{2}-9x^{2}}$$

Since \frac{3x+2y}{2\left(3x-2y\right)\left(3x+2y\right)} and \frac{r\left(3x-2y\right)}{2\left(3x-2y\right)\left(3x+2y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x+2y-r\left(3x-2y\right)}{2\left(3x-2y\right)\left(3x+2y\right)}+\frac{3x}{4y^{2}-9x^{2}}$$

Do the multiplications in 3x+2y-r\left(3x-2y\right).

$$\frac{3x+2y-3rx+2ry}{2\left(3x-2y\right)\left(3x+2y\right)}+\frac{3x}{4y^{2}-9x^{2}}$$

Factor 4y^{2}-9x^{2}.

$$\frac{3x+2y-3rx+2ry}{2\left(3x-2y\right)\left(3x+2y\right)}+\frac{3x}{\left(-3x+2y\right)\left(3x+2y\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(3x-2y\right)\left(3x+2y\right) and \left(-3x+2y\right)\left(3x+2y\right) is 2\left(-3x+2y\right)\left(3x+2y\right). Multiply \frac{3x+2y-3rx+2ry}{2\left(3x-2y\right)\left(3x+2y\right)} times \frac{-1}{-1}. Multiply \frac{3x}{\left(-3x+2y\right)\left(3x+2y\right)} times \frac{2}{2}.

$$\frac{-\left(3x+2y-3rx+2ry\right)}{2\left(-3x+2y\right)\left(3x+2y\right)}+\frac{2\times 3x}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Since \frac{-\left(3x+2y-3rx+2ry\right)}{2\left(-3x+2y\right)\left(3x+2y\right)} and \frac{2\times 3x}{2\left(-3x+2y\right)\left(3x+2y\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(3x+2y-3rx+2ry\right)+2\times 3x}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Do the multiplications in -\left(3x+2y-3rx+2ry\right)+2\times 3x.

$$\frac{-3x-2y+3rx-2ry+6x}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Combine like terms in -3x-2y+3rx-2ry+6x.

$$\frac{3x-2y+3rx-2ry}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Factor the expressions that are not already factored in \frac{3x-2y+3rx-2ry}{2\left(-3x+2y\right)\left(3x+2y\right)}.

$$\frac{\left(r+1\right)\left(3x-2y\right)}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Extract the negative sign in 3x-2y.

$$\frac{-\left(r+1\right)\left(-3x+2y\right)}{2\left(-3x+2y\right)\left(3x+2y\right)}$$

Cancel out -3x+2y in both numerator and denominator.

$$\frac{-\left(r+1\right)}{2\left(3x+2y\right)}$$

Expand 2\left(3x+2y\right).

$$\frac{-\left(r+1\right)}{6x+4y}$$

To find the opposite of r+1, find the opposite of each term.

$$\frac{-r-1}{6x+4y}$$

Factor

$$-\frac{r+1}{2\left(3x+2y\right)}$$