Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of multiplication to write an equivalent algebraic expression. $$7 x+23$$

Step-by-Step Solution

Answer

The equivalent algebraic expression using the commutative property of multiplication is \(x \times 7 + 23\)

Step 1: Identify the expression

The given expression is \(7 x+23\). We can see that it contains the multiplication operation \(7 \times x\).

Step 2: Apply the commutative property

We can apply the commutative property of multiplication to \(7 \times x\). According to this property, \(7 \times x\) can be written as \(x \times 7\).

Step 3: Write the equivalent expression

After applying the commutative property to \(7 \times x\), the equivalent expression is \(x \times 7 + 23\)

Key Concepts

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators combined uniquely in mathematical phrases. They include components like terms, coefficients, and constants. In the expression \(7x + 23\), \(7x\) is a term where 7 is the coefficient multiplied by the variable \(x\). The number 23 is a constant added to the term.

• Terms: Each part of the expression separated by a plus or minus sign.
• Coefficients: The number multiplied by the variable(s) in a term.
• Constants: Numbers without variables, standing alone in the expression.

Understanding each component is crucial, as it helps manipulate and simplify algebraic expressions using different mathematical properties.

Mathematical Properties

Mathematical properties are fundamental rules that apply to numbers and operations. These properties help in simplifying, rearranging, and solving expressions or equations. One crucial property in algebra is the Commutative Property.

• The Commutative Property of Addition states that changing the order of numbers in an addition operation does not change the sum, i.e., \(a + b = b + a\).
• The Commutative Property of Multiplication follows a similar rule; flipping the order of factors does not change the product, i.e., \(a \times b = b \times a\).

These properties are key tools when manipulating algebraic expressions, enabling expressions to be rewritten in different yet equivalent forms. This is particularly helpful when simplifying or solving algebraic equations.

Multiplication in Algebra

In algebra, multiplication is a vital operation used to solve equations and simplify expressions. When multiplying a number by a variable, you are scaling the variable. For instance, in the term \(7x\), the number 7 acts as a scaling factor for the variable \(x\).

• Variable Multiplication: A variable multiplied by a number adds that repeated number's amount to the total or expression.
• Using Properties: Applying properties like the Commutative Property aids in restructuring terms. For \(7x\), rewriting it as \(x \times 7\) doesn't change its value but demonstrates flexibility in expression representations.

Understanding multiplication in algebra helps solve equations more effectively and efficiently by allowing you to move terms and factors into more convenient positions within expressions.