Problem 7 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{x}{3}-\frac{1}{3}$$

Step-by-Step Solution

Answer

The result is \(\frac{x-1}{3}\).

Step 1: Understand the Problem

The given problem is to subtract two fractions: \(\frac{x}{3}\) and \(\frac{1}{3}\). Both fractions have the same denominator, which simplifies the subtraction process.

Step 2: Subtract the Fractions

Since the fractions have the same denominator, subtract the numerators directly: \(x - 1\). The denominator remains the same, resulting in \(\frac{x - 1}{3}\).

Step 3: Reduce to Lowest Terms

Check if the expression \(\frac{x-1}{3}\) can be simplified further. Here, \(x-1\) and 3 have no common factors, therefore the fraction is already in its simplest form.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In the context of fractions like \( \frac{x}{3} \) and \( \frac{1}{3} \), the expression involves a variable \( x \) which makes it an algebraic expression.

Variables in algebraic expressions represent unknown values that can change or be defined by specific values. The flexibility of variables allows us to perform arithmetic operations and simplify expressions. This helps in solving algebraic equations and understanding relationships between variables. For example, in the expression \( x-1 \), the variable \( x \) can be any real number which defines the entire algebraic expression.
- Variables are symbols that can vary.- Expressions can be simplified but not solved unless given an equation.- Algebraic expressions can include fractions, constants, and variables.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms by finding common factors in the numerator and the denominator. This makes fractions easier to work with and understand. In our exercise, the fraction result of the subtraction is \( \frac{x-1}{3} \).

To simplify a fraction like \( \frac{x-1}{3} \), we check for any possible values that could simplify the expression:- Identify common factors in the numerator and denominator.- Simplify by dividing both by their greatest common factor (GCF).

For instance, if \( x = 4 \), \( x-1 = 3 \) and the expression becomes \( \frac{3}{3} \). By simplifying, you get 1, which is the simplest form of the fraction. If there are no common factors, as in our original subtraction problem, then the fraction is already simplified.

Like Denominators

When fractions have like denominators, or the same denominator, it simplifies both the addition and subtraction operations. In our problem, the fractions \( \frac{x}{3} \) and \( \frac{1}{3} \) both have 3 as their denominator.

Having a common denominator means we can directly subtract or add the numerators without needing to find a common base first. This makes the operation straightforward:- Keep the denominator, \( d \), the same.- Perform the operation on the numerators alone.
For example, in the subtraction from our exercise, you subtract the numerators: \( x - 1 \), while keeping 3 as the fixed denominator. This results in \( \frac{x-1}{3} \), making the process efficient without needing additional steps to adjust the denominators.