Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(2(x+3)=2 x+6\)

Step-by-Step Solution

Answer

The algebraic property illustrated by the statement \(2(x+3)=2x+6\) is the Distributive Property.

Step 1: Identify the Algebraic Property

The equation is: \(2(x+3)=2x+6\). Here multiplication is distributed over addition, which is an operation inside the parentheses.

Step 2: Apply the Distributive Property

The left-hand side is the term before simplification and the right-hand side is the term after simplification, according to the distributive property which states that \(a(b+c)=ab+ac\). Here, \(a=2, b=x, c=3\). So, \(2*(x+3)\) should equal to \(2*x + 2*3\).

Step 3: Final Verification

In this case, \(2*(x+3)\) is equal to \(2*x + 6\), which is same as the right-side of the equation \(2x+6\). Therefore, the equation illustrates the Distributive Property.

Key Concepts

Algebraic Properties

In algebra, properties are rules that apply to numbers and mathematical operations. One essential property is the **Distributive Property**. This property allows you to break down expressions and tackle them in a simplified manner.
In any algebraic expression, the distributive property is represented as:

• For numbers: \( a(b + c) = ab + ac \)
• This means you can distribute the multiplication over the terms inside the brackets.

This rule extends beyond just numbers to include variables, making it a powerful tool in algebraic calculations and manipulations.

Simplification in Algebra

Simplification in algebra refers to the process of making an expression easier to read and work with. It's all about reducing complexity by combining like terms or using properties like the distributive property.
When dealing with expressions such as \(2(x + 3)\), applying the distributive property makes it easier to see hidden relationships between terms:

• The multiplication of 2 across the terms inside the parentheses results in \(2 \cdot x + 2 \cdot 3\).
• This simplifies the expression to \(2x + 6\).

Simplification decreases chances for errors and makes solving equations much faster, acting as a crucial step towards clearer solutions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations. They represent quantities and relationships in algebra. In the expression \(2(x + 3)\), there is a mix of:

• Variables \(x\) which denote unknowns you often solve for.
• Numbers such as 2 and 3 which are constants known as coefficients or constants.

Expressions provide a precise way to describe mathematical situations or problems, serving as the backbone of algebra. Understanding how to manipulate and interpret these expressions is critical for solving equations.

Equation Solving

The process of equation solving involves finding the value(s) for the variable(s) that make the equation true. For example, with the equation \(2x + 6 = 12\), solving involves isolating the variable:

• You start by subtracting 6 from both sides: \(2x = 6\).
• Then divide by 2 to find \(x = 3\).

Using properties like the distributive property, as seen with \(2(x+3)=2x+6\), makes identifying relationships faster. Solving equations accurately relies on understanding and properly applying algebraic rules and simplification techniques to reach correct solutions efficiently.