simplify (1/y+1/x)/(x/y-y/x) — step-by-step solution

Let's simplify:

\frac{\frac{1}{y} + \frac{1}{x}}{\frac{x}{y} - \frac{y}{x}}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{y} + \frac{1}{x}

\frac{x + y}{y x}

\frac{x}{y} - \frac{y}{x}

\frac{x \cdot x - y \cdot y}{y x}

\frac{\frac{x + y}{y x}}{\frac{x \cdot x - y \cdot y}{y x}}

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

y \cdot y

y^{2}

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

ⓘ Dividing is multiplying by the reciprocal:

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

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

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

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

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

\frac{(x + y) \cdot y x}{y x \cdot (x^{2} - y^{2})}

\frac{(x + y) \cdot y x}{y x \cdot (x^{2} - y^{2})}

Using the formula:

a^{2} - b^{2} = (a - b) \cdot (a + b)

, we simplify the fraction:

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

by the common factor:

x + y

to get the new fraction:

\frac{1}{x - y}

▶Simplified form:

\frac{1}{x - y}