Problem 57 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ (9 a+6 b-5)+(-11 a-7 b+6) $$

Step-by-Step Solution

Answer

The simplified expression is \(-2a - b + 1\).

Step 1: Identify Like Terms

In the expression \((9a + 6b - 5) + (-11a - 7b + 6)\), identify the like terms. These are terms that have the same variable to the same power.- Like terms for \(a\): \(9a\) and \(-11a\)- Like terms for \(b\): \(6b\) and \(-7b\)- Constant like terms: \(-5\) and \(6\)

Step 2: Group Like Terms

Rewrite the expression by grouping the like terms together:\[ (9a - 11a) + (6b - 7b) + (-5 + 6) \]

Step 3: Apply Addition/Subtraction to Like Terms

Now, simplify each group of like terms by performing the addition or subtraction:- For \(a\) terms: \(9a - 11a = -2a\)- For \(b\) terms: \(6b - 7b = -b\)- For the constants: \(-5 + 6 = 1\)

Step 4: Write the Simplified Expression

Combine the simplified terms to write the final expression:\[ -2a - b + 1 \]

Key Concepts

Like Terms

When it comes to polynomial addition and subtraction, understanding 'Like Terms' is crucial. Like terms are terms within an expression that share the same variable(s) raised to the same power. Let's look at a simple example. Consider the expression

• Example:
  9a + 6b - 5
• Another Expression:
  -11a - 7b + 6

In the given expression \( (9a + 6b - 5) + (-11a - 7b + 6) \), the terms that are "like" share both the variable and the degree. This means:

• The terms 9a and -11a are both considered like terms because they involve the variable 'a'.
• Similarly, 6b and -7b are like terms since they both contain the variable 'b'.
• The constants -5 and 6 are also considered like terms because they are just numbers without any variables.

Once we identify the like terms, we can simplify the expression by grouping and combining them to solve the equation efficiently.

Expression Simplification

To simplify an algebraic expression, such as in polynomial expressions, the goal is to rewrite it in the simplest form. Simplification makes expressions easier to manage and calculate.Start by identifying like terms, then combine them through addition or subtraction. When you encounter an expression like\((9a + 6b - 5) + (-11a - 7b + 6)\), follow these steps:

• Combine like terms for 'a': 9a - 11a results in -2a.
• Combine like terms for 'b': 6b - 7b results in -b.
• Finally, add the constants: -5 + 6 equals 1.

After gathering the simplified results, combine them into a single cohesive expression: \(-2a - b + 1\).Through simplification, complex expressions are reduced into smaller, equivalent expressions that are easier to work with and understand.

Grouping Terms

Grouping terms is vital for managing and simplifying polynomial expressions. By physically rearranging an expression into logical segments, solving becomes clearer and more intuitive.Let's take the expression we have been discussing: \( (9a + 6b - 5) + (-11a - 7b + 6) \). Initially, it may seem daunting, but by grouping terms based on their similarities, we simplify the handling:

• For terms involving 'a', we regroup to see: \((9a - 11a)\).
• For terms involving 'b', the arrangement becomes: \((6b - 7b)\).
• Finally, align the constant terms: \((-5 + 6)\).

The act of rearranging terms into grouped forms allows us to focus on one type of term at a time, facilitating a step-by-step simplification approach.Grouping is a crucial step in ensuring each part of the polynomial is systematically simplified, ultimately guiding you to a concise and neat solution.