Problem 2 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$x(3)=3(x)$$

Step-by-Step Solution

Answer

This equation is justified by the commutative property of multiplication.

Step 1: Understanding the Equation

The equation given is \(x(3) = 3(x)\). Here, both sides involve multiplication of two elements: \(x\) and \(3\). We are tasked with identifying the property that allows these two expressions to be equivalent.

Step 2: Identifying the Property

The property at play here is the commutative property of multiplication. This property states that for any two numbers \(a\) and \(b\), the equation \(a \cdot b = b \cdot a\) holds true. This is because the order in which you multiply numbers does not affect the product.

Step 3: Applying the Property to the Equation

In our example, \(x\) takes the place of \(a\) and \(3\) takes the place of \(b\) in the commutative property of multiplication. Therefore, \(x(3) = 3(x)\) holds true because the multiplication is commutative.

Key Concepts

Properties of Numbers

Numbers have fascinating properties that help us solve algebraic expressions more efficiently. One crucial property is the commutative property of multiplication. This property states that the order in which you multiply numbers does not matter. For instance, when you multiply two numbers, like 5 and 7, you can switch them around:

• 5 \( \times \) 7 = 35
• 7 \( \times \) 5 = 35

In both cases, you end up with the same result. This exchangeable nature of numbers makes calculations flexible and easier to handle in expressions and equations. When you see similar expressions in algebra, like \( x \cdot 3 = 3 \cdot x \), know that you are looking at a perfect example of the commutative property at work.

Equality of Expressions

Understanding equality in mathematical expressions is key to solving equations effectively. An expression is a combination of numbers, variables, and operations that represent a particular value. The commutative property helps establish the equality of expressions:

• When two expressions have equal values, they are equal expressions.
• For example, in the expression \( x \cdot 3 = 3 \cdot x \), both are equal because they result in the same outcome no matter the value of \( x \).

Knowing expressions are equal lets you confidently move forward in solving equations, ensuring each side represents the same quantity. This keeps your algebraic work balanced and accurate, preventing errors in calculations.

Algebraic Equations

Algebraic equations are like a balanced scale, with both sides representing the same amount. Solving these equations often requires understanding properties of numbers, like the commutative property. An equation like \( x \cdot 3 = 3 \cdot x \) shows how properties of numbers maintain balance:

• Both sides have the same value because of the commutative property of multiplication.
• This ensures the equation holds true for any real number \( x \).

To solve any algebraic equation, always aim to maintain equality. Use properties such as the commutative property to rearrange terms and balance both sides. These strategies help you efficiently find solutions and understand the underlying concepts with ease.