Problem 22 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3(x+y)-2(x-y) $$

Step-by-Step Solution

Answer

The simplified expression is \( x + 5y \).

Step 1: Distribute Each Term

Start by distributing the constants outside the parentheses to each term inside. This means multiplying both terms inside the parentheses by the number outside. For the first part, distribute 3 into the terms inside the parentheses: \[ 3(x+y) = 3 \cdot x + 3 \cdot y = 3x + 3y \]Similarly, for the second part, distribute -2 into the terms inside the parentheses:\[ -2(x-y) = -2 \cdot x + (-2) \cdot (-y) = -2x + 2y \]

Step 2: Write Expanded Expression

Using the results from the previous step, write the expanded form of the original expression. Substitute the distributed expressions:\[ 3x + 3y - 2x + 2y \]

Step 3: Group Similar Terms

Identify and group similar terms. Similar terms are those with the same variable or constant term:\[ (3x - 2x) + (3y + 2y) \]

Step 4: Combine Similar Terms

Combine the coefficients of the grouped terms to simplify the expression:\[ (3x - 2x) = 1x = x \]\[ (3y + 2y) = 5y \]Therefore, the simplified expression is:\[ x + 5y \]

Key Concepts

Simplifying Expressions

Simplifying an algebraic expression is all about making it as basic as possible. Essentially, we want to rewrite the expression in a way that looks tidier, without changing its value. This process involves distributing terms, finding like terms, and combining them to come up with a version that’s cleaner and easier to comprehend.

In essence:

• Distribute terms across parentheses.
• Identify like terms with similar components.
• Combine these terms to reflect a simpler form.

This process helps us solve algebraic equations more efficiently by focusing only on the essential parts of the expression. It's like arranging clutter in your room so you can clearly see and easily access everything you need.

Distributive Property

The distributive property is a key principle in algebra that allows us to simplify expressions effectively. It states that when you multiply a number by a sum contained within parentheses, you distribute the multiplication to each term inside those parentheses.

For example, in the expression \( 3(x+y) \), the distributive property allows us to multiply 3 by both \( x \) and \( y \). Thus, it becomes: \( 3 \times x + 3 \times y = 3x + 3y \).

Similarly, any negative number outside parentheses should also be distributed inside. So, consider the expression \(-2(x-y)\). Here, distribute -2 to both \( x \) and \( -y \), leading to \(-2x + 2y\).

• The distributive property breaks down complex expressions, making them easier to handle.
• It simplifies the process of dealing with parentheses in algebraic expressions.

Understanding this property sets a foundation for tackling more complicated equations with confidence.

Combining Like Terms

Combining like terms is a fundamental algebraic concept that allows you to simplify expressions by merging terms that have the same variable component. This step is crucial because it reduces the clutter of dealing with multiple similar terms and brings out the essential structure of the expression.

For instance, once you have distributed terms such as \( 3x + 3y - 2x + 2y \), the next step involves combining like terms.

Recognize terms that share the same variable, such as \( 3x \) and \( -2x \), and merge them together: \( (3x - 2x) = x \). Similarly, group \( 3y \) and \( 2y \) together: \( (3y + 2y) = 5y \).

Tips for combining like terms:

• Focus on coefficients attached to the same variable.
• Reorganize terms logically to keep the extension straightforward.
• Ensure all terms related to the same variable are merged.

By effectively combining like terms, you streamline the expression, making calculation and analysis much more manageable.