Solve 3y/4x-2y-y/x-1/2

$$\frac{ 3y }{ 4x-2y } - \frac{ y }{ x } - \frac{ 1 }{ 2 }$$

Evaluate

$$-\frac{x^{2}-y^{2}}{x\left(2x-y\right)}$$

Short Solution Steps

Factor 4x-2y.

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

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

$$\frac{3yx}{2x\left(2x-y\right)}-\frac{y\times 2\left(2x-y\right)}{2x\left(2x-y\right)}-\frac{1}{2}$$

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

$$\frac{3yx-y\times 2\left(2x-y\right)}{2x\left(2x-y\right)}-\frac{1}{2}$$

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

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

Combine like terms in 3yx-4yx+2y^{2}.

$$\frac{-yx+2y^{2}}{2x\left(2x-y\right)}-\frac{1}{2}$$

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

$$\frac{-yx+2y^{2}}{2x\left(2x-y\right)}-\frac{x\left(2x-y\right)}{2x\left(2x-y\right)}$$

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

$$\frac{-yx+2y^{2}-x\left(2x-y\right)}{2x\left(2x-y\right)}$$

Do the multiplications in -yx+2y^{2}-x\left(2x-y\right).

$$\frac{-yx+2y^{2}-2x^{2}+xy}{2x\left(2x-y\right)}$$

Combine like terms in -yx+2y^{2}-2x^{2}+xy.

$$\frac{-2x^{2}+2y^{2}}{2x\left(2x-y\right)}$$

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

$$\frac{2\left(x-y\right)\left(-x-y\right)}{2x\left(2x-y\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(x-y\right)\left(-x-y\right)}{x\left(2x-y\right)}$$

Expand x\left(2x-y\right).

$$\frac{\left(x-y\right)\left(-x-y\right)}{2x^{2}-xy}$$

Apply the distributive property by multiplying each term of x-y by each term of -x-y.

$$\frac{-x^{2}-xy+yx+y^{2}}{2x^{2}-xy}$$

Combine -xy and yx to get 0.

$$\frac{-x^{2}+y^{2}}{2x^{2}-xy}$$

Factor

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