Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of addition to write an equivalent algebraic expression. $$4+9 x$$

Step-by-Step Solution

Answer

The equivalent algebraic expression for \(4+9x\) using the commutative property of addition is \(9x + 4\).

Step 1: Identify the terms

The given expression is \(4 + 9x\). Here, the terms are 4 and \(9x\). The commutative property of addition can be applied to these terms.

Step 2: Apply the Commutative Property

Using the commutative property of addition \(a + b = b + a\), the equivalent algebraic expression can be written by swapping the terms 4 and \(9x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For example, the expression 4 + 9x involves the addition of the number 4 and the term 9x, which is the product of 9 and the variable x. Understanding how these expressions are structured is crucial for manipulating and simplifying them using various algebraic properties.

An important aspect of algebraic expressions is that they can represent quantities in abstract terms, allowing us to solve problems without knowing the exact values of the variables. The ability to work with these expressions builds the foundation for more complex mathematical concepts, such as functions and equations.

Breaking Down the Parts

Every algebraic expression consists of terms, which are the separated elements that are added or subtracted. In 4 + 9x, there are two terms: the constant 4 and the variable term 9x. Constants stand alone, while variable terms include both a coefficient (in this case, 9) and a variable (x), representing unknown quantities.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that represent the same value for all values of the variables involved. This concept is critical for simplifying expressions and solving equations. By using properties of algebra, it is possible to transform expressions into simpler forms while keeping their values unchanged.

For the given expression 4 + 9x, one can apply the commutative property of addition to rearrange the terms, resulting in an equivalent expression, 9x + 4. These expressions are equivalent because, no matter what value is substituted for the variable x, they will both yield the same result.

Verifying Equivalence

To verify the equivalence of two expressions, one could evaluate both expressions for several values of the variable(s) and compare the results. If they match for all tested values, the expressions are equivalent. In algebraic contexts, properties like the commutative, associative, and distributive properties are often used to demonstrate the equivalence of expressions.

Properties of Addition

The properties of addition are rules that describe how numbers behave when they are added together. These properties are fundamental in arithmetic and algebra and can greatly simplify calculations and the manipulation of expressions.

The commutative property of addition states that the order in which two numbers are added does not affect the sum, i.e., a + b = b + a. It allows for the flexibility of rearranging terms in an expression, which is particularly useful in algebra for simplifying expressions and solving for variables.

Other Addition Properties

Aside from the commutative property, there are other important properties of addition to note:

• The associative property ((a + b) + c = a + (b + c)) implies that the grouping of addends does not affect the sum.
• The identity property of addition (a + 0 = a) indicates that adding zero to any number does not change its value.
• The additive inverse property (a + (-a) = 0) suggests that any number added to its negative results in zero.

These properties collectively enable us to perform arithmetic efficiently and to lay the groundwork for solving algebraic equations.