Problem 59 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{2 x+2 y}{3} \div \frac{x^{2}-y^{2}}{x-y}$$

Step-by-Step Solution

Answer

The result of the given division is \(\frac{2}{3}\). Remember that simplification after factorization played a major role in coming up with this solution

Step 1: Rewrite Division as Multiplication

The division of fractions can be rewritten as the multiplication by the reciprocal of the second fraction. Hence, the original problem \(\frac{2 x+2 y}{3} \div \frac{x^{2}-y^{2}}{x-y}\) becomes \(\frac{2 x+2 y}{3} \times \frac{x-y}{x^{2}-y^{2}}\)

Step 2: Factorise

The expression \(2x + 2y\) can be factorised as \(2(x+y)\), and the expression \(x^2 - y^2\) can be factorised as \((x-y)(x+y)\) using the difference of squares formula. Now, the problem becomes \(\frac{2(x+y)}{3} \times \frac{x-y}{(x-y)(x+y)} \)

Step 3: Simplify the Result

Now, we notice that \((x-y)\) is a common factor in the numerator and the denominator, which can be canceled. After simplification, the problem becomes \(\frac{2(x+y)}{3(x+y)}\)

Step 4: Further Simplification

Again, notice that \((x+y)\) is a common factor in the numerator and the denominator, that can be canceled. After simplification, we get the final answer \(\frac{2}{3}\)