Problem 33 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 5 x y-3 y $$

Step-by-Step Solution

Answer

The simplified expression is \(y(5x - 3)\).

Step 1: Identify the Terms in the Expression

The expression consists of two terms: \(5xy\) and \(-3y\). Each term is a product of coefficients and variables, specifically involving \(x\) and \(y\).

Step 2: Identify Common Factors

Examine the terms \(5xy\) and \(-3y\) to identify common factors. Both terms share the variable \(y\) as a common factor.

Step 3: Factor Out the Greatest Common Factor (GCF)

Factor \(y\) out of each term in the expression. When factoring \(y\) from \(5xy\), you get \(5x\). When factoring \(y\) from \(-3y\), you get \(-3\). The expression becomes \(y(5x - 3)\).

Step 4: Write the Simplified Expression

After factoring out the greatest common factor, the simplified expression is \(y(5x - 3)\). This is the most reduced form of the given expression.

Key Concepts

Factoring

Factoring is like "unpacking" an expression. We look for numbers or variables that are common in the terms and "factor" them out. This means we rewrite the expression as a product of simpler parts. Think of it like finding the biggest box that each part can fit into. Factoring is an essential tool for simplifying expressions and solving equations.

• Look at each term to find what they have in common—these can be numbers, variables, or both.
• Rewrite the expression so that it includes this common factor as part of a multiplication outside of a parenthesis.
• This method creates a more compact and often more understandable form of the expression.

In this task, we had the expression made up of two terms: \(5xy\) and \(-3y\). We recognized a shared factor in both, which is what allows for factoring.

Greatest Common Factor

The greatest common factor (GCF) is like the largest box that can fit parts of each element in the expression. Identifying the GCF is crucial in factoring expressions efficiently. When you find the GCF, you have the largest possible number or variable that can divide each term without leaving a remainder.

• Examine each term carefully to find its GCF. Look for the highest degree of each variable present in all terms.
• Finding the GCF involves identifying the maximum "commonness" between the terms.
• The GCF can include numbers, variables, or a combination of both.

In the example provided, the terms \(5xy\) and \(-3y\) both contain the variable \(y\). So, the GCF here is \(y\). Factoring out \(y\) simplifies the expression significantly.

Simplification

Simplifying an expression means making it as straightforward and compact as possible. It often involves factoring and reducing expressions to a simpler form. Simplification not only makes it easier to understand but also easier to work with the expression in equations or larger calculations.

• Once the GCF is factored out, you can rewrite the expression in its simplest form.
• Look at the resulting terms inside the parenthesis. These should be the simplest forms of each term after factoring.
• Simplifying makes it easier to study the properties of the expression, such as finding roots or graphing it.

By factoring \(y\) from \(5xy - 3y\), the expression was simplified to \(y(5x - 3)\). This form is easier to read and can be used to quickly identify needed information in more complex problems.