Question

The fact that \(2(x+3)=2 x+6\) is attributable to the _________ Property.

Short answer

Distributive Property

Step-by-step solution

Understand the Equation

Examine the equation provided: \(2(x+3)=2x+6\).Notice that the left-hand side is a product of two and a sum, and the right-hand side is the result of distributing the two into the terms inside the parentheses.

Apply the Correct Property

Identify the property used to go from \(2(x+3)\) to \(2x + 6\). This involves multiplying each term inside the parentheses by 2.

Recognize the Property

Realize that the property applied here is the Distributive Property, which states that for any numbers a, b, and c, \(a(b + c) = ab + ac\).

Key concepts

Algebraic Properties

Algebraic properties are rules that help us manipulate mathematical expressions and equations.
They are foundational concepts in algebra that ensure our calculations are accurate.
Understanding these properties enables us to simplify complex expressions and solve equations more easily.
Some key algebraic properties include:

• Commutative Property: The order of addition or multiplication does not change the result. For example, 2 + 3 = 3 + 2 and 4 x 5 = 5 x 4.
• Associative Property: The way numbers are grouped in addition or multiplication does not change the result. For example, (2 + 3) + 4 = 2 + (3 + 4).
• Distributive Property: This property allows us to distribute a multiplication over addition or subtraction, which we will explore in detail in the next section.

Knowing these properties helps simplify and solve algebraic equations efficiently.

Distributive Property

The Distributive Property is a fundamental algebraic property used to simplify expressions.
It states that for any numbers a, b, and c:
\[ a(b + c) = ab + ac \]
This means we multiply each term inside the parentheses by the number outside.
For example, in the equation provided, \(2(x + 3)\):

• First, distribute the 2 to both x and 3.
• This results in \(2 \times x + 2 \times 3\).
• Which simplifies to \(2x + 6\).

So, \(2(x + 3) = 2x + 6\), demonstrating the Distributive Property in action.
This property is incredibly useful for simplifying expressions and solving equations involving parentheses.
By understanding and applying the Distributive Property, you can break down more complex equations into simpler, more manageable parts.

Equation Solving

Solving equations in algebra involves finding the value of an unknown variable that makes the equation true.
Generally, we use algebraic properties to manipulate the equation until the variable is isolated.
Here’s a simple approach to solve equations using the Distributive Property:

• First, apply the Distributive Property to eliminate parentheses, if any.
• Next, combine like terms on both sides of the equation.
• Then, isolate the variable by performing inverse operations (e.g., addition, subtraction, multiplication, and division).
• Finally, solve for the variable.

Using our example \(2(x + 3) = 2x + 6\), we see that the equation is already simplified using the Distributive Property.
If we needed to isolate x further, we could use inverse operations accordingly.
Mastering these steps is critical for solving more complex algebraic equations.

Multiplication

Multiplication is one of the basic arithmetic operations that we use extensively in algebra.
It involves finding the product of two numbers or terms.
When dealing with algebraic expressions, multiplication helps us distribute terms and simplify equations.
For example, in the Distributive Property \(a(b + c)\), we multiply a by both b and c.
In our example, \(2(x + 3)\), multiplication is used as follows:

• Multiply 2 by x to get \(2x\).
• Multiply 2 by 3 to get 6.

The result \(2x + 6\) shows how multiplication distributes across addition inside the parentheses.
Understanding how to correctly multiply terms within and outside parentheses is key to mastering algebra.
It ensures that you can simplify and solve equations accurately, maintaining the integrity of the mathematical expressions.