Problem 62 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{2(y - x)}{x^{2}-y^{2}}\).

Step 1: Recognize the Property

Notice that the denominators are each other's opposites. Specifically, \(x^{2}-y^{2}\) and \(y^{2}-x^{2}\) are additive inverses, which means that multiplying them would give \( -1 \). Thus, it can be rewritten as: \(\frac{2 y}{x^{2}-y^{2}} - \frac{2 x}{x^{2}-y^{2}}\).

Step 2: Combine Like Terms

The fractions have the same denominator now, that allows us to combine the fractions by adding or subtracting the numerators, as shown below:\(\frac{2 y - 2 x}{x^{2}-y^{2}}\).

Step 3: Simplify the Fraction

Now, the numerator can be simplified taking common out as: \(\frac{2(y - x)}{x^{2}-y^{2}}.\)

Key Concepts

Additive Inverses

In the world of mathematics and algebra, the concept of additive inverses is fundamental. It's the idea that every number has an opposite, also known as an additive inverse, which when added together, equal zero. This is crucial when dealing with algebraic fractions because recognizing additive inverses can help simplify expressions.
In our exercise involving rational expressions, we have the denominators \(x^2 - y^2\) and \(y^2 - x^2\). These expressions are additive inverses of each other. Simply put, their sum is zero: \ x^2 - y^2 + y^2 - x^2 = 0.\ This is because substituting one in place of the other results in \(-1\), making your calculations tidier. Recognizing these inverses allows you to rewrite the expression to more simply solve the problem.

Rational Expressions

Rational expressions are essentially fractions where the numerator and/or the denominator comprise polynomials. These expressions behave much like regular fractions, but they involve variables. Understanding rational expressions is important since they regularly appear in algebra and pre-calculus.
In the original exercise, two rational expressions are given:\(\frac{2y}{x^2-y^2}\) and \(\frac{2x}{y^2-x^2}\). These fractions involve polynomials in the denominator, making them rational expressions. When dealing with these kinds of expressions, pay attention to the denominators so you can find common factors or simplify as needed.
Rational expressions often require being combined, typically through addition or subtraction, thus finding a common denominator is key, which in this case, was leveraging the additive inverses.

Simplifying Fractions

Once you've combined fractions, the next step usually involves simplifying them. Simplifying fractions is a way of breaking them down into their simplest form, making calculations and future operations easier.
In the final step of the solution, we simplify the expression \(\frac{2(y-x)}{x^2-y^2}\). Because we know the polynomial \(x^2-y^2\) can be rewritten using the difference of squares as \ (x-y)(x+y), \ understanding factorization becomes beneficial here. We can't simplify further in this particular case due to \(y-x\) in the numerator being relatively prime to \(x-y\) in the denominator, but recognizing these patterns can help in similar exercises.
Simplifying fractions often involves factorization, cancelling out common terms, and knowing your algebraic identities! Keep practicing to spot these chances quickly in various problems.