Problem 2 - Algebra 1: Concepts and Skills

Question

Explain how you would use the distributive property to simplify the expression. $$ (x+4) 5 $$

Step-by-Step Solution

Answer

The simplified form of the expression \((x + 4) \cdot 5\) is \(5x + 20\) using the distributive property.

Step 1: Understand the Distributive Property

The Distributive Property of Multiplication over Addition states that for any numbers \(a\), \(b\), and \(c\), the equation \(a \cdot (b + c) = a \cdot b + a \cdot c\) holds. This property allows us to expand expressions and eliminate parentheses.

Step 2: Apply the Distributive Property

In this case, the expression can be viewed as \(5 \cdot (x + 4)\). Treat \(5\) as \(a\), \(x\) as \(b\) and \(4\) as \(c\) using the distributive property. Therefore, \(5 \cdot (x + 4)\) can be expanded to \(5 \cdot x + 5 \cdot 4\).

Step 3: Simplify the Expression Further

Simplify the expression further by performing the multiplication: \(5 \cdot x + 5 \cdot 4\) simplifies to \(5x + 20\).

Key Concepts

Expression Simplification

Expression simplification means rewriting an expression in a simpler form without changing its value. It often involves reducing the number of terms and making the expression easier to understand.
In algebra, simplifying expressions is crucial because it allows us to see relationships between variables and constants more clearly.
When simplifying an expression using the distributive property, you expand it by removing parentheses and combining like terms if possible.
For example, in the expression \((x + 4) \cdot 5\), applying the distributive property helps simplify it to \(5x + 20\), making further operations easier to handle.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not have an equal sign, which distinguishes them from equations.
An algebraic expression might look like \(3x + 2\), \(a^2 - 4b + c\), or even something like \(y/2 + 6\).
These expressions can get pretty complex, so knowing how to simplify them, like using the distributive property, helps keep them manageable.

• Terms in an expression: Parts of an expression separated by addition or subtraction signs, e.g., in \(2y + 3\), "2y" and "3" are terms.
• Coefficients: Numbers multiplied by variable terms, e.g., in \(2y\), "2" is a coefficient.
• Constants: Numbers on their own in an expression, e.g., "3" in \(2y + 3\).

Understanding these components is essential in manipulating and simplifying algebraic expressions.

Multiplication Over Addition

Multiplication over addition is a key part of the distributive property. It shows how multiplying a sum by a number affects each term within the sum individually.
For example, consider the expression \(a \cdot (b + c)\). Using multiplication over addition, you would multiply both \(b\) and \(c\) by \(a\), resulting in \(a \cdot b + a \cdot c\).
In the exercise \((x + 4) \cdot 5\), recognize \(5\) as the multiplier. You distribute \(5\) across the terms inside the parentheses: both \(x\) and \(4\). This process simplifies the expression to \(5x + 20\).
This method is a powerful tool in simplifying complex expressions and solving algebraic equations, making them easier to interpret and solve.