Problem 44 - Elementary Algebra

Question

Simplify. $$ -6 a+7 \beta-2 a+\beta $$

Step-by-Step Solution

Answer

\(-8a + 8\beta\)

Step 1: Group Like Terms

Identify and group like terms from the expression \(-6a + 7\beta - 2a + \beta\).The like terms are:- Terms with \(a\): \(-6a - 2a\)- Terms with \(\beta\): \(7\beta + \beta\)

Step 2: Simplify Terms with 'a'

Simplify the expression containing \(a\):Combine \(-6a\) and \(-2a\):\[-6a - 2a = -8a\]

Step 3: Simplify Terms with 'β'

Simplify the expression containing \(\beta\):Combine \(7\beta\) and \(\beta\):\[7\beta + \beta = 8\beta\]

Step 4: Combine Simplified Terms

Combine the simplified terms from previous steps to get the final expression:\(-8a + 8\beta\)

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. It's important to understand like terms because they are the cornerstone of simplifying algebraic expressions. In the exercise, terms such as \( -6a \) and \( -2a \) are like terms because they both include the variable \( a \) raised to the first power. Similarly, \( 7\beta \) and \( \beta \) have the same variable \( \beta \).

Recognizing like terms makes simplification much easier, as it allows us to combine them through addition or subtraction. Without identifying them correctly, simplification could lead to mistakes.

• Like terms must have the exact same variable component.
• Only the coefficients (the numbers in front of the variables) of like terms are combined, not the variables.

Understanding this concept will help you in solving more complex algebraic expressions with confidence.

Grouping Terms

Grouping terms is a technique used to organize like terms before performing any mathematical operations on them. This step ensures that we only combine terms that are indeed compatible due to sharing the same variable and exponent.

In the original exercise, different terms were grouped as follows:

• \( -6a - 2a \)
• \( 7\beta + \beta \)

Grouping like terms is like sorting clothes in a laundry. Only after sorting can you wash them with the proper method according to their type.

Once grouped, these terms can then be easily combined. This helps maintain accuracy and simplifies the calculations needed to obtain the final expression.

Algebraic Simplification

Algebraic simplification involves reducing an algebraic expression to its simplest form. This is done by combining the like terms after grouping them. It's essentially tidying up an expression to make it as concise as possible while still retaining its value.

Looking at the exercise, after grouping, the expression \( -6a + 7\beta - 2a + \beta \) was simplified by combining the grouped terms into:

• \( -8a \) from \( -6a - 2a \)
• \( 8\beta \) from \( 7\beta + \beta \)

This led to the final simplified expression: \( -8a + 8\beta \).

Simplification helps in making expressions easier to read and solve, especially in more complicated equations or when solving for unknown variables. Remember:

• Always perform arithmetic operations on coefficients of like terms while keeping the variable part unchanged.
• The goal is to have the simplest expression possible which is easier to work with in later mathematical or real-world applications.