Problem 82 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{y-x y}{x y-x} $$

Step-by-Step Solution

Answer

The expression cannot be simplified further and remains \(\frac{y(1-x)}{x(y-1)}\).

Step 1: Simplify the Numerator and Denominator

The given expression is \(\frac{y - xy}{xy - x}\). First, let's try to simplify both the numerator \(y - xy\) and the denominator \(xy - x\). Both terms have a common factor: the numerator can be written as \(y(1 - x)\) and the denominator as \(x(y - 1)\). So the expression becomes \(\frac{y(1-x)}{x(y-1)}\).

Step 2: Factor Common Terms

Notice that in the numerator \(y(1-x)\) and the denominator \(x(y-1)\), neither expression has common factors that are identical, since \(1-x\) and \(y-1\) are not the same. Therefore, no further simplification is possible here.

Step 3: Verifying Simplification

We check if further simplification can be done by ensuring that no common factors exist aside from what was previously identified. As \(1-x\) and \(y-1\) are distinct and cannot cancel each other out, the expression \(\frac{y(1-x)}{x(y-1)}\) is in its simplest form.

Key Concepts

Factoring Expressions

Factoring expressions is like breaking down a complicated problem into smaller, simpler parts. When you factor an expression, you’re looking to express it as a product of its components, or 'factors'.

For example, if we have the expression \(y - xy\), what we can do is find what these two terms have in common. In this case, both terms have a common factor of \(y\). If we factor \(y\) out of the expression, it becomes \(y(1-x)\).

Similarly, for the denominator \(xy - x\), the common factor is \(x\). By factoring \(x\), we rewrite the expression as \(x(y-1)\).

Once factored, expressions often reveal simpler forms that are easier to work with. Factoring is a fundamental step in processes such as simplifying algebraic fractions.

Common Factors

Understanding common factors can dramatically simplify algebraic expressions. A common factor is a shared number or variable that divides two or more numbers or terms evenly.

In our exercise, the numerator \(y - xy\) and the denominator \(xy - x\) both initially appear complex. However, identifying their common factors—\(y\) for the numerator and \(x\) for the denominator—allows for a straightforward simplification process.

Once you identify these, you pull them out of their respective expressions. This technique not only simplifies the expression but also makes further operations, like addition or division, more manageable. Recognizing common factors is critical in algebra, as it often reveals the simplest form of expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. They behave much like regular fractions but with variables.

In our exercise, the rational expression \(\frac{y - xy}{xy - x}\) is dealt with by simplifying both the numerator and the denominator through factoring. After factoring out common factors, we end up with \(\frac{y(1-x)}{x(y-1)}\).

One key aspect when working with rational expressions is to attempt simplification by canceling common factors in both the numerator and the denominator. However, as shown, \(1-x\) and \(y-1\) don't match, so no further simplification through canceling is possible.

Handling rational expressions involves understanding both the factoring and the properties of polynomials, which aids in managing complex algebraic fractions.