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Chapter 7: Problem 59

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

Short Answer

Expert verified
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 by step solution

01

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 \(\frac{2 x+2 y}{3} \times \frac{x-y}{x^{2}-y^{2}}\)
02

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 \(\frac{2(x+y)}{3} \times \frac{x-y}{(x-y)(x+y)} \)
03

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 \(\frac{2(x+y)}{3(x+y)}\)
04

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 \(\frac{2}{3}\)

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

Will help you prepare for the material covered in the next section. If \(B=k W,\) find the value of \(k,\) in decimal form, using \(B=5\) and \(W=160\)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-8}{x^{2}-16}-\frac{x-8}{16-x^{2}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{11}{x+7}-\frac{5}{-x-7}$$

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)
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