Problem 69 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ -7 b^{2}+27 b^{2} $$

Step-by-Step Solution

Answer

20b^2

Step 1: Identify Like Terms

First, identify the like terms in the expression \[-7b^2 + 27b^2\]. Like terms are terms that have the same variables raised to the same powers. In this expression, both terms \(-7b^2\) and \(27b^2\) are like terms because they both contain \(b^2\).

Step 2: Combine Like Terms

Since the terms are like terms, we can combine them by adding their coefficients together. The coefficients are \(-7\) and \(27\), so we perform the addition: \(-7 + 27 = 20\). Therefore, the simplified expression becomes \(20b^2\).

Key Concepts

Identifying Like Terms

To simplify an algebraic expression, we first need to identify the like terms. Recognizing like terms is key because they can be combined to make the equation simpler.

**What are Like Terms?**

• Like terms have the same variable parts.
• These variables must be raised to the same power.
• For example, in the expression \(-7b^2 + 27b^2\), both \(-7b^2\) and \(27b^2\) contain the same variable, \(b\), raised to the power of 2.
• Unlike terms would have different variables or powers, like \((-7b^2)\) and \(27b\).

Identifying like terms helps in organizing and simplifying algebraic expressions effectively. Always look for these commonalities to know what can be combined.

Coefficients

Coefficients are the numeric parts of a term. Understanding coefficients can help you combine like terms easily.

**Understanding Coefficients**

• The coefficient is the number directly in front of the variable in a term.
• In the expression \(-7b^2 + 27b^2\), the terms \(-7b^2\) and \(27b^2\) are like because they have \(b^2\) with coefficients \(-7\) and \(27\) respectively.
• When terms are like terms, add the coefficients to combine them, while the variable part remains unchanged.

By focusing on adding or subtracting these numeric parts, coefficients are essential for effectively simplifying expressions.

Simplifying Expressions

Once you've identified like terms, the next step is simplifying the expression by combining them.

**Simplification Process**

• Identify all sets of like terms in the expression.
• Combine each set by adding or subtracting their coefficients.
• For \(-7b^2 + 27b^2\), calculate \(-7 + 27\) which results in \(20\).
• Attach the common variable part, \(b^2\), to get the simplified expression \(20b^2\).

By breaking down the expression into steps, you simplify it piece by piece. Simplifying makes the expression easier to understand and work with in further mathematical operations.