Problem 72 - Elementary Algebra

Question

Use the distributive property to expand \(3(x+6)\).

Step-by-Step Solution

Answer

Question: Expand the expression \(3(x+6)\) using the distributive property. Answer: The expanded form of the expression is \(3x + 18\).

Step 1: Apply the distributive property

Using the distributive property, we can rewrite the expression \(3(x+6)\) as \(3\cdot x + 3 \cdot 6\).

Step 2: Calculate the product

Now, we can find the product of each term: \((3\cdot x) + (3\cdot 6) = 3x + 18\).

Step 3: Write the final expression

The expanded form of the given expression using the distributive property is: \(3x + 18\).

Key Concepts

Expanding Algebraic Expressions

Expanding algebraic expressions is a fundamental skill in algebra. It involves breaking down a group of terms that are initially in a compact form, usually within parentheses, and expressing them in a more spread out form.
This process often uses the distributive property to simplify complex equations. In the expression \(3(x+6)\), expansion requires distributing the "\(3\)" across each term inside the parentheses.
By applying this method, we rewrite expressions in a more understandable way, making it easier to perform further operations like addition or subtraction.

• Breakdown: Identify components that need to be multiplied.
• Rewriting: Distribute each term.
• Resulting expression: Show the expanded form.

Understanding and mastering the ability to expand expressions helps build a robust foundation for more advanced algebraic techniques.

Step-by-Step Algebra Solutions

Understanding algebra becomes easier when you break it into simple, manageable steps. A step-by-step approach helps in not getting overwhelmed by the complexity of algebraic expressions.
You start by clearly identifying each element that needs to be solved or simplified. Then, one step at a time, you solve the parts until reaching the final expression.
For the expression \(3(x+6)\), the first step was to apply the distributive property. By multiplying "\(3\)" with each term inside, you obtain \(3 \cdot x + 3 \cdot 6\).
Let's highlight why dividing the solution into simpler steps is crucial:

• Clarity: Steps make the solution transparent and easy to follow.
• Structure: Each part is individually tackled.
• Confidence: Build your understanding block by block.

By practicing this approach, you gain the confidence and ability to solve more complicated expressions with ease.

Multiplying Expressions

Multiplying expressions is a key activity in algebra that combines terms from different algebraic expressions to form new expressions. It involves distributing each number or variable across other numbers or variables.
For instance, in the problem \(3(x+6)\), the multiplication involves taking "\(3\)" and applying it to both \(x\) and \(6\). This technique results in \(3x + 18\).
To master multiplication in algebraic contexts, remember to:

• Identify: Find which terms need to be multiplied.
• Distribute: Apply multiplication to each part of the expression independently.
• Simplify: Combine any like terms for tidy results.

Getting comfortable with these processes allows you to tackle more complex problems effortlessly, and creates a strong grasp on the fundamentals of algebra.