Solve ((x - y)/x - (y - x)/y) / ((x + y)/(xy)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(\frac{x-y}{x}-\frac{y-x}{y}):\frac{x+y}{xy}$$

Evaluate

$x-y$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x$ and $y$ is $xy$. Multiply $\frac{x-y}{x}$ times $\frac{y}{y}$. Multiply $\frac{y-x}{y}$ times $\frac{x}{x}$.

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

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

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

Do the multiplications in $\left(x-y\right)y-\left(y-x\right)x$.

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

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

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

Divide $\frac{-y^{2}+x^{2}}{xy}$ by $\frac{x+y}{xy}$ by multiplying $\frac{-y^{2}+x^{2}}{xy}$ by the reciprocal of $\frac{x+y}{xy}$.

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

Cancel out $xy$ in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in $-x-y$.

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

Cancel out $x+y$ in both numerator and denominator.

$$-\left(-x+y\right)$$

Expand the expression.

$$x-y$$

Expand

$x-y$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x$ and $y$ is $xy$. Multiply $\frac{x-y}{x}$ times $\frac{y}{y}$. Multiply $\frac{y-x}{y}$ times $\frac{x}{x}$.

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

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

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

Do the multiplications in $\left(x-y\right)y-\left(y-x\right)x$.

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

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

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

Divide $\frac{-y^{2}+x^{2}}{xy}$ by $\frac{x+y}{xy}$ by multiplying $\frac{-y^{2}+x^{2}}{xy}$ by the reciprocal of $\frac{x+y}{xy}$.

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

Cancel out $xy$ in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in $-x-y$.

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

Cancel out $x+y$ in both numerator and denominator.

$$-\left(-x+y\right)$$

Expand the expression.

$$x-y$$