Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The property used is the Distributive Property.

Step 1: Identify the Expression

The given expression is \((5x + 1) \cdot 3 = 15x + 3\). We need to identify what operation is performed between \((5x + 1)\) and \(3\).

Step 2: Distribute the Coefficient

Apply the Distributive Property of real numbers to the left-hand side of the equation: \[(5x + 1) \cdot 3 = 5x \cdot 3 + 1 \cdot 3\]

Step 3: Calculate the Result

Calculate each term from the distributive step: \[5x \cdot 3 = 15x\] and \[1 \cdot 3 = 3\].Therefore, we have \[15x + 3\].

Step 4: Compare Both Sides

The expression on the left simplifies to \[15x + 3\], which is identical to the right-hand side \[15x + 3\]. This confirms that the operation used is correct and the property applied is appropriate.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing both rational and irrational numbers. Rational numbers include fractions and whole numbers, while irrational numbers cannot be expressed as fractions. Real numbers are crucial because they help us represent quantities in daily life, such as temperature, distance, and time.

Understanding real numbers lays the groundwork for comprehending more advanced mathematical properties and operations. In our example, the expression \((5x + 1) \cdot 3 = 15x + 3\) involves real numbers. Here, 5, 1, and 3 are coefficients and constants, which are all real numbers. These numbers multiply with the variable \(x\) and with each other, demonstrating how real numbers interact in expressions. The goal is to simplify the relationships between these numbers using mathematical properties.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They serve as the language of algebra, allowing us to model and solve real-world problems. For example, in the expression \((5x + 1) \cdot 3 = 15x + 3\), we see a typical algebraic setup.

An algebraic expression contains different parts, such as:

• Coefficients: The numbers multiplying the variables, like 5 in \(5x\)
• Variables: Symbols that represent unknown values, such as \(x\)
• Constants: Fixed numbers that don't change, like 1 in the expression
• Operations: The mathematical actions applied to the numbers and variables, such as addition (+) and multiplication (\(\cdot\))

In solving algebraic expressions, we often simplify them or perform operations to make calculations easier. Using properties like the distributive property helps make these expressions clearer and shows how variables and constants interact.

Multiplication Property

The multiplication property in mathematics states how numbers and variables multiply, impacting the structure and solution of algebraic expressions. The distributive property is a specific instance of this, showing how to distribute a single multiplier across terms within parentheses.

For example, the expression \((5x + 1) \cdot 3\) uses the distributive property to simplify as follows:

• Multiply \(5x\) by 3 to get \(15x\)
• Multiply 1 by 3 to get 3

This yields the simplified form \(15x + 3\). The distributive property ensures that multiplication is correctly performed on each term within the parentheses, maintaining equality in an equation. Understanding this property is vital for working with algebraic expressions and solving equations efficiently.