Problem 11 - Precalculus Mathematics for Calculus

Question

State the property of real numbers being used. $$(5 x+1) 3=15 x+3$$

Step-by-Step Solution

Answer

Distributive Property

Step 1: Identify the Property

The equation is \[(5x+1)3=15x+3\]This is the distributive property where a number outside the parentheses is multiplied by each term inside the parentheses.

Step 2: Rewrite Using the Distributive Property

Let's apply the distributive property: \[3(5x) + 3(1) = 15x + 3\]Simplifying, it becomes: \[15x + 3 = 15x + 3\]

Key Concepts

Properties of Real Numbers

The properties of real numbers are fundamental rules that govern operations involving real numbers. Understanding these properties is crucial as they form the foundation for various algebraic manipulations and problem-solving techniques. The main properties include:

• Commutative Property: This property applies to both addition and multiplication. For example, for any two numbers, \(a + b = b + a\) and \(a \times b = b \times a\).
• Associative Property: This property also applies to addition and multiplication. It states that the grouping of numbers does not affect the result, i.e., \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: This is a key property where multiplication is distributed over addition or subtraction, shown as \(a(b + c) = ab + ac\).
• Identity Property: For addition, the identity is 0 since \(a + 0 = a\). For multiplication, the identity is 1 since \(a \times 1 = a\).
• Inverse Property: For any real number, the additive inverse is \(-a\) so that \(a + (-a) = 0\). The multiplicative inverse is \(\frac{1}{a}\) for any non-zero \(a\), such that \(a \times \frac{1}{a} = 1\).

The exercise above utilizes the distributive property, which is essential for solving algebraic expressions.

Algebraic Expressions

Algebraic expressions are compositions of numbers, variables, and arithmetic operators. They are used to represent mathematical relationships or to model real-world situations. A typical algebraic expression involves one or more terms with operations like addition, subtraction, multiplication, and division.
In the original exercise, the expression \((5x + 1)3\) demonstrates how algebra can model operations with variables. Here, the number 3 is multiplied across each term of the expression inside the parentheses, showcasing the distributive property. This transforms the expression into \(15x + 3\). It illustrates how expressions can be manipulated into simpler forms, which is crucial for solving equations or performing further operations.
A deep understanding of algebraic expressions entails:

• Identifying terms and coefficients
• Understanding the role of variables as placeholders for numbers
• Recognizing polynomials, which are expressions consisting of multiple terms
• Applying operations correctly to simplify or evaluate expressions

Grasping these concepts allows learners to transition into solving more complex equations effectively.

Equation Solving

Solving equations is the process of finding the values of variables that satisfy the equation. It involves applying algebraic techniques and properties to isolate the variable on one side of the equation.
In the provided solution, after applying the distributive property, the expression simplifies to \(15x + 3 = 15x + 3\), which is an identity. This means that the equation holds true for any value of \(x\), illustrating an important aspect of equation solving where sometimes operations lead to identities or contradictions instead of specific solutions.
Key steps in equation solving include:

• Identification: Recognizing the form and type of equation, whether linear, quadratic, etc.
• Manipulation: Using properties like the distributive property to simplify or rearrange the equation.
• Isolation: Moving all terms with the variable of interest to one side while consolidating constants on the other side.
• Verification: Substituting the solution back into the original equation to ensure correctness.

A thorough understanding of these steps equips students with the skills necessary to tackle a diverse range of mathematical problems.