Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Use an associative property to rewrite each algebraic expression. Once the grouping has been changed, simplify the resulting algebraic expression. $$7+(5+x)$$

Step-by-Step Solution

Answer

The resulting simplified expression after applying the associative property is \(12 + x\).

Step 1: Apply the associative property

Applying the associative property to \(7+(5+x)\), the expression can be regrouped as \((7+5)+x\).

Step 2: Simplify the expression

Simplify the expression \((7+5)+x\) as \(12 + x\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a cornerstone of algebra and are essentially a combination of numbers, variables (like x), and operation signs (such as +, -, *, /). They're an attempt to capture mathematical relationships in a general form. Take, for example, the expression 7+(5+x). It represents a quantity with a known part, 7 + 5, and an unknown part, x. Algebraic expressions can become as complex as necessary to model real-world scenarios, but even the most intricate expressions rely on basic principles, like those you're learning.

Understanding these expressions is crucial, as they're used in various applications such as solving equations, evaluating functions, and modeling real-life situations. As you encounter algebraic expressions, remember they are not just mathematical puzzles but tools to solve problems.

Simplifying Expressions

Simplifying expressions is like tidying up a cluttered room, making it easier to see and understand what's there. The primary goal is to write the expression in its simplest form, removing parentheses and combining like terms when possible. Simplification is a handy skill that makes other algebra tasks, like solving equations or working with functions, much more manageable.

In our example (7+5)+x, the simplification involves adding the numbers within the parentheses first. Why? Because it's easier to deal with a single number (12 in this case) than an addition operation when moving forward. When you simplify expressions, always look for opportunities to combine or reduce elements – sometimes, like in our example, it only takes one step, but other times it may require several. Just remember, the simpler, the better!

Properties of Addition

The properties of addition—which include the associative, commutative, and identity properties—are fundamental rules that describe how numbers behave when we add them together. The associative property, in particular, tells us that the grouping of numbers does not affect their sum. This means that (a + b) + c is the same as a + (b + c).

In the context of our exercise, we used the associative property to change the grouping from 7 + (5 + x) to (7 + 5) + x, but the sum remains unchanged. This property is incredibly helpful for simplifying expressions, allowing flexibility in how we approach calculations. It not only simplifies the math we do on paper but also hints at the broader concept of commutative operations in mathematical structures like groups and fields, which are foundational to more advanced mathematics.