Problem 31 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{2 x-10}{3 x-30} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{2(x-5)}{3(x-10)}\).

Step 1: Factor the Numerator

First, look at the numerator of the expression, which is \(2x - 10\). Notice that both terms have a common factor of \(2\). Factor \(2\) out to get \(2(x - 5)\).

Step 2: Factor the Denominator

Next, factor the denominator \(3x - 30\). Like the numerator, both terms share a common factor. Here, it is \(3\). Factor \(3\) out to get \(3(x - 10)\).

Step 3: Simplify the Expression

Rewrite the fraction using the factored forms: \[\frac{2(x - 5)}{3(x - 10)}\]. Since there are no common factors in the numerator and the denominator, this is the simplified form of the expression.

Key Concepts

Factoring Expressions

Factoring expressions is a handy technique that involves breaking down an expression into simpler components. Think of it as finding what to multiply together to form an original expression. Let's consider the expression from the exercise: \(2x - 10\).
First, identify the common factor among the terms. In this case, both 2 and 10 can be divided by 2. Thus, we pull out or "factor out" this number, resulting in \(2(x - 5)\).
The same goes for the denominator \(3x - 30\). By recognizing 3 as a common factor, we can transform it into \(3(x - 10)\).
Factoring expressions can make complex fractions simpler and reveal opportunities for further simplification later on.

Simplification Techniques

Simplification in algebraic fractions often involves reducing the expression to its simplest form. This can be achieved by canceling or removing any common factors that appear in both the numerator and denominator.
In our given exercise, we factored the numerator and denominator to \(2(x - 5)\) and \(3(x - 10)\) respectively. At first glance, remember to check if anything can be canceled out between these factored expressions. However, since there are no common terms left between the top and bottom, the expression is already in its simplest form.
Being thorough when simplifying helps ensure accuracy, and it's always a good practice to double-check your work for possible further simplification steps.

Common Factors

Identifying common factors is a crucial step when dealing with expressions and fractions. Common factors are numbers or variables that divide into each term of an expression without leaving a remainder.
For example, in the numerator \(2x - 10\), the common factor is 2. This means both 2 and 10 divide evenly by 2. Similarly, in the denominator \(3x - 30\), the common factor is 3.
Recognizing these factors helps you rewrite expressions in a "factored form," which can then be useful for simplifying the entire fraction. Once factored, check for shared factors across fractions to further reduce them. This is a fundamental technique for simplified fraction representations.