Problem 7 - Elementary Algebra

Question

Reduce each of the following fractions to lowest terms. $$ \frac{2 x-y}{y-2 x} $$

Step-by-Step Solution

Answer

Question: Reduce the fraction $$\frac{2x-y}{y-2x}$$ to its lowest terms. Answer: -1

Step 1: Identify common factors in the numerator and denominator

In the given fraction, we have $$\frac{2x-y}{y-2x}.$$ Notice that both the numerator and the denominator have common factors. Let's factor out these common terms.

Step 2: Swap the order in the denominator

We can swap the terms in the denominator, which will have the effect of changing the signs. We now have: $$\frac{2x-y}{-(2x-y)}.$$

Step 3: Simplify the fraction

Now, we notice that the numerator and the denominator are same except for the signs. This can be simplified further as follows: $$\frac{2x-y}{-(2x-y)} = -\frac{2x-y}{2x-y} = -1.$$ Now the fraction has been reduced to its lowest terms, which is -1 in this case.

Key Concepts

Reducing Fractions

Reducing fractions is the process of simplifying a fraction to its simplest form. Imagine you have a fraction, and you want to make it as easy as possible to understand or use. That’s what reducing does. It simplifies the fraction by dividing both the numerator and the denominator by their greatest common factor (GCF).
In the context of algebraic fractions, this can involve many steps, often beginning with rewriting the expression in a way that reveals common factors. By dividing these common factors out of both the top and bottom, you make the fraction as simple as possible.
For example, with algebraic expressions like \(\frac{2x-y}{y-2x}\), recognizing patterns or using factoring strategies helps find common factors, which simplifies the expression quickly. In this instance, swapping terms or recognizing the negative factor helped us reduce the fraction.

Factoring

Factoring is the method used to simplify expressions by breaking them down into ‘factors’. Factors are expressions or numbers you multiply together to get the original expression.
In algebraic contexts, factoring is vital for simplifying, solving equations, and reducing fractions. When you factor an expression, you are looking for expressions that multiply together to recreate it. Sometimes this involves recognizing patterns, such as differences of squares or common terms to factor out.
In the exercise \(\frac{2x-y}{y-2x}\), factoring involves recognizing that swapping the order of terms can change the expression into \(\frac{2x-y}{-(2x-y)}\). Notice how the change in sign is critical here. It’s like uncovering a hidden puzzle, where recognizing a common factor lends power to simplifying the equation.

Simplifying Expressions

Simplifying expressions is akin to tidying up mathematical expressions. You aim to represent them in the most basic or streamlined form.
This involves reducing complexity in terms of how many operations a given expression contains, or rewriting fractions to their simplest terms. The process requires a combination of identifying common factors, using basic arithmetic manipulations, and understanding properties of operations like addition, subtraction, multiplication, and division.
In our example \(\frac{2x-y}{-(2x-y)}\), once the common factor and structure was identified, simplifying it was straightforward. It resulted in reducing the entire expression to \(-1\), showing that even complex algebraic fractions can often condense to simple, elegant conclusions.