Problem 3 - Elementary Algebra

Question

Simplify. $$ x y-3 y $$

Step-by-Step Solution

Answer

The simplified expression is \(y(x - 3)\).

Step 1: Identify Common Term

Look at each term in the expression \(xy - 3y\). Identify the common factor in both terms. In this case, both terms have \(y\) as a common factor.

Step 2: Factor Out the Common Term

After identifying \(y\) as the common factor, factor it out from both terms. This means we pull \(y\) in front of a parentheses. The expression \(xy - 3y\) is rewritten as \(y(x - 3)\).

Step 3: Verify the Factoring

To ensure the factoring is correct, distribute \(y\) back into the expression inside the parentheses: \(y(x - 3) = y \cdot x - y \cdot 3\) which simplifies back to \(xy - 3y\). This confirms our factorization is accurate.

Key Concepts

Common Factor

In algebra, a common factor is an element that appears in each term of an algebraic expression. For the expression \(xy - 3y\), the common factor is \(y\) because it is present in both terms: \(xy\) and \(-3y\).
Identifying common factors is a crucial first step in the process of simplifying expressions.

• It helps identify components of terms that can be simplified.
• Spotting a common factor allows us to factor this out of the equation, which can make equations easier to handle or further simplify.

Finding common factors is much like looking for common elements in a group – once found, they can be treated as a single unit and moved outside the equation for a cleaner form.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. This often means factoring expressions to reveal simpler, equivalent expressions. Take the expression \(xy - 3y\).
Once we identify and factor out the common factor \(y\):

• We rewrite the expression as \(y(x - 3)\).
• This rearranges the original expression to highlight its inherent structure.

Simplification helps in several ways:

• Makes complex expressions more manageable.
• Reveals underlying patterns or relationships between terms.
• Is a useful technique for solving equations, graphing, and performing calculus operations.

In essence, simplification is about making expressions friendlier to work with by focusing on their core components.

Distributive Property

The distributive property is a foundational algebraic principle that shows how multiplication interacts with addition or subtraction. It states that if you have a term outside a parenthesis that multiplies an addition or subtraction inside, you distribute the multiplication across all terms inside.
For example, in verifying the expression \(y(x - 3)\), you use the distributive property:

• Multiply \(y\) by each term inside the parentheses: \(y \cdot x\) and \(-y \cdot 3\).
• Resulting in the expansion: \(yx - 3y\).
• This shows that \(y(x - 3)\) simplifies back to the original \(xy - 3y\).

Utilizing the distributive property:

• Ensures transformations or simplifications you've made maintain initial values.
• Aids in cross-verifying the accuracy of factored expressions.
• Is pivotal in legal manipulations of equations across diverse fields of math.

Understanding and applying this property boosts the effectiveness of algebraic manipulation skills.