Chapter 11.2, Problem 68PPE

To determine

Divide the expressions: $\frac{5x^2}{y^2-25}÷\frac{5xy-25x}{y^2-10y+25}$

Answer to Problem 68PPE

  $\frac{5x^2}{y^2-25}÷\frac{5xy-25x}{y^2-10y+25}=\frac{x}{y+5}$

Explanation of Solution

Given:

Divide the expressions: $\frac{5x^2}{y^2-25}÷\frac{5xy-25x}{y^2-10y+25}$

Concept Used:

Find the factors of numerator and denominator, and then cancel the common factor and simplify.

For division of rational expression, we always flip the 2^nd expression and put a multiplication sign.

Rule: $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

Calculation:

  $\begin{array}{l}\frac{5x^2}{y^2-25}÷\frac{5xy-25x}{y^2-10y+25}\\ \\ \frac{5x^2}{y^2-25}·\frac{y^2-10y+25}{5xy-25x}\\ \\ \frac{5·x·x}{(y+5)(y-5)}·\frac{(y-5)(y-5)}{5·x(y-5)}\\ \\ \frac{y-5}{y-5}·\frac{y-5}{y-5}·\frac{5·x}{5·x}·\frac{x}{y+5}=\frac{x}{y+5}\end{array}$

Thus, $\frac{5x^2}{y^2-25}÷\frac{5xy-25x}{y^2-10y+25}=\frac{x}{y+5}$