Problem 23 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{x+7}{7+x} $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Analyze the Expression

Look at the expression \(\frac{x+7}{7+x}\). Notice that the numerator is \(x+7\) and the denominator is \(7+x\).

Step 2: Identify a Possible Simplification

Notice that \(x+7\) and \(7+x\) are the same after reordering the terms, because addition is commutative, i.e., \(a + b = b + a\).

Step 3: Simplify the Expression

Since \(x + 7 = 7 + x\), replace \(7 + x\) in the denominator with \(x + 7\). The expression becomes \(\frac{x+7}{x+7}\).

Step 4: Final Simplification

Cancel the common terms \((x+7)\) in the numerator and the denominator, which results in the simplified expression 1.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the expression \(\frac{x+7}{7+x}\), both the numerator and the denominator are algebraic expressions. The numerator is \(x+7\) which consists of the variable \(x\) and the number 7 added together. Similarly, the denominator is \(7+x\), which also combines a number and a variable. Algebraic expressions can be simplified by applying mathematical rules and properties. These include operations like addition, subtraction, multiplication, division, distribution, and factoring. Simplification aims to make the expression easier to work with or solve.

Commutative Property

The commutative property is a fundamental property of addition and multiplication. It states that changing the order of the numbers or variables does not change the result. In mathematical terms, if \(a\) and \(b\) are two numbers or expressions, then:

• For addition: \(a + b = b + a\)
• For multiplication: \(a \cdot b = b \cdot a\)

This property is crucial in simplification because it allows us to reorder terms in expressions without changing their values. In the expression \(x + 7\) and \(7 + x\), applying the commutative property helps us recognize that they are essentially the same expression, facilitating further simplification steps.

Numerator and Denominator

In a fraction, the numerator and denominator are the two parts that define its value. The fraction \(\frac{x+7}{7+x}\) represents the division of the numerator \(x+7\) by the denominator \(7+x\). The role of the numerator is to signify how many parts of a whole are being considered, whereas the denominator indicates the total number of equal parts in that whole. Understanding the distinction between these two parts is important when dealing with fractions and simplifying them. Recognizing that both parts of our fraction are equivalent expressions due to the commutative property allows for simplification.

Cancellation

Cancellation is a technique used to simplify fractions when the same term appears in both the numerator and the denominator. When the terms are identical, they "cancel each other out" because any non-zero number divided by itself equals one. For instance, in the expression \(\frac{x+7}{x+7}\), the term \(x+7\) is both in the numerator and the denominator. By canceling, you simplify the expression to 1. This step is valid as long as the common term being canceled is not zero, ensuring the fraction remains defined. Understanding cancellation is essential for simplifying complex fractions and can often make algebraic expressions much simpler.