Problem 68 - Algebra 2

Question

Simplify each expression. $$ 6 a-2 b-3 a+9 b $$

Step-by-Step Solution

Answer

The simplified expression is \(3a + 7b\).

Step 1: Identify Like Terms

First, observe the expression to identify like terms. Like terms are terms that contain the same variable raised to the same power. In the expression \(6a - 2b - 3a + 9b\), the like terms are \(6a\) and \(-3a\) (terms involving \(a\)), and \(-2b\) and \(9b\) (terms involving \(b\)).

Step 2: Group Like Terms Together

Rearrange the expression by grouping like terms together. This helps in simplifying them easily. The expression \(6a - 2b - 3a + 9b\) becomes \((6a - 3a) + (-2b + 9b)\).

Step 3: Simplify Each Group of Like Terms

Now, simplify each group of like terms separately. For the \(a\) terms, calculate \(6a - 3a = 3a\). For the \(b\) terms, calculate \(-2b + 9b = 7b\).

Step 4: Combine Simplified Results

Combine the simplified results from each group to obtain the simplified expression. The expression becomes \(3a + 7b\).

Key Concepts

Like Terms

Understanding the concept of like terms is crucial in algebraic simplification. Like terms are those that contain the same variables raised to the same power. For example, in our expression, both \(6a\) and \(-3a\) are like terms because they both have the variable \(a\), raised to the first power. Similarly, \(-2b\) and \(9b\) are like terms since they also share the variable \(b\), again raised to the same power.
Recognizing like terms allows us to add or subtract them, just like regular numbers. This is a powerful technique because it simplifies the expression and makes it easier to work with.
Here are tips to identify like terms in any expression:

• Look at each term and identify the variables and their exponents.
• Group terms with matching variables and exponents.
• Remember, terms like \(3a^2\) and \(5a\) are not like terms, because their exponents differ.

Expression Simplification

Expression simplification is the process of making an algebraic expression more concise and easier to understand. By simplifying an expression, you reduce it to a form that is easier to work with and interpret.
This can be achieved through various algebraic techniques, one of which is combining like terms. We take the original expression \(6a - 2b - 3a + 9b\), and begin by reorganizing it into groups based on like terms, which reflects the student's understanding of addition and subtraction in algebra.
Expression simplification is crucial because:

• It reduces the complexity of mathematical operations.
• It provides clarity, especially when dealing with large equations or performing further calculations.
• Simplified expressions are easier to evaluate and solve.

Grouping terms by common variables and reducing them significantly condenses the expression's length and complexity.

Combining Like Terms

Combining like terms is one of the foundational skills in algebra, which involves adding or subtracting coefficients of terms that have the same variable parts. It is crucial for simplifying expressions and solving equations more efficiently.
In the given expression, combining the \(a\) terms \(6a - 3a = 3a\), and the \(b\) terms \(-2b + 9b = 7b\), leads to a significantly simplified expression \(3a + 7b\).

Steps to combine like terms effectively:

• Identify like terms in the expression.
• Reorganize the expression so that like terms are next to each other.
• Add or subtract the coefficients of these like terms.
• Rewrite the expression with the simplified terms.

The process of combining like terms not only simplifies the expression but also enhances your ability to manipulate more complex equations accurately.