Problem 10 - Intermediate Algebra

Question

Fill in the blanks to simplify \(\frac{x-y}{y-x}\) $$ \frac{x-y}{y-x}=\frac{-y+}{y-x}=\frac{(y-x)}{(y-x)}= $$

Step-by-Step Solution

Answer

The simplified expression is -1.

Step 1: Recognize the Negative

The expression in the numerator, \(x-y\), can be rewritten by factoring out a negative sign. Notice that \(x-y = -(y-x)\). This will help us simplify the fraction.

Step 2: Rewrite the Fraction

Replace \(x-y\) with \(-(y-x)\) in the fraction to get \(\frac{-(y-x)}{y-x}\). This takes advantage of the common terms in the numerator and denominator.

Step 3: Simplify the Fraction

Since \(y-x\) appears in both the numerator (with a negative sign) and the denominator, they can be canceled out. This leaves us with \(-1\), since dividing any non-zero number by itself equals 1, and the negative sign remains.

Key Concepts

Factoring

Factoring is a process of breaking down expressions into simpler pieces that can easily work together to solve or simplify the mathematical problem. These pieces are called "factors," and they can be numbers, variables, or even whole expressions. Factoring helps identify equivalent expressions that can make the problem clearer.

In the original exercise, we started by factoring out a negative sign. This means rewriting an expression in a way that highlights the "negative" relationship between terms. For example, given the expression \(x-y\), we can see it as \( -(y-x) \). By factoring, we made use of a property where multiplying by a negative number inverts the order of subtraction. Recognizing this reverse order as a negative factor is key in simplification problems.

Negative Numbers

Negative numbers are numbers less than zero and are often used to denote opposite directions or quantities in mathematics. Dealing with negative numbers includes understanding their properties when added, subtracted, multiplied, or divided.

In the expression \(x-y = -(y-x)\), we use the property of negative numbers where changing the order of subtraction brings a negative sign in front. This is a fundamental concept that aids in simplifying expressions where the same terms appear in both the numerator and the denominator, allowing us to effectively "cancel" them out as shown in the solution.

Fractions

Fractions represent a part of a whole and are a foundational concept in math where one integer is divided by another. The fraction \(\frac{x-y}{y-x}\) showcases how two expressions can represent proportionate values of each other.

Fractions can often be simplified by finding common factors in the numerator and the denominator. When both the top (numerator) and bottom (denominator) of a fraction have the same terms, they can be reduced or canceled, simplifying the fraction to a simpler form. In our exercise, identifying that \(x-y\) could be rewritten and factored as \(-1(y-x)\) allowed us to simplify the fraction dramatically.

Numerator and Denominator

The numerator and denominator are the two main parts of a fraction. The numerator is on top and indicates how many parts we have, while the denominator, at the bottom, tells us into how many parts the whole is divided.

In the fraction \(\frac{x-y}{y-x}\), \(x-y\) serves as the numerator, and \(y-x\) is the denominator. To simplify such fractions effectively, we need to think about the commonalities between these two parts. By factoring negative signs properly, we can identify when terms in the numerator can be directly paired with those in the denominator, simplifying calculations and expressing the relationship between the numerator and denominator more clearly, which in this case simplifies to \(-1\).