Problem 44 - Algebra A Combined Function

Question

Simplify the expressions. $$ (9 q+p)+(-9 q-p) $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Identify Like Terms

First, look at the expression \((9q + p) + (-9q - p)\). Notice that there are terms with the variable \(q\) and terms with the variable \(p\) that are similar and can be combined.

Step 2: Group Like Terms

Group together the terms containing \(q\) and the terms containing \(p\). The expression now looks like this: \((9q) + (-9q) + (p) + (-p)\).

Step 3: Combine Like Terms

Simplify each group of like terms separately. The terms \((9q) + (-9q)\) combine to 0, and \((p) + (-p)\) also combine to 0.

Step 4: Write the Simplified Expression

After combining all like terms, the entire expression simplifies to 0. Therefore, the expression \((9q + p) + (-9q - p)\) simplifies to 0.

Key Concepts

Understanding Like Terms

In algebra, the topic of **like terms** is essential for simplifying expressions. Like terms are those that contain the same variables raised to the same power. They can be easily identified by looking at the variables and exponents.

• Example: In the expression \(9q + p + (-9q) - p\), notice the terms \(9q\) and \(-9q\) both contain the variable \(q\).
• Another example: The terms \(p\) and \(-p\) share the variable \(p\).

Recognizing like terms is the first step in simplifying expressions. Once identified, these terms can be combined to make calculations more straightforward. Understanding and identifying like terms lays the groundwork for successful algebraic manipulation.

Combining Like Terms

After identifying like terms, the next important step is **combining** them. This process involves adding or subtracting the coefficients (the numerical part) of the like terms. By doing this, you can simplify expressions effectively.

• For example, combining \(9q + (-9q)\) involves adding the coefficients: \(9\) and \(-9\). Thus, \(9q + (-9q) = 0\).
• Similarly, combining \(p + (-p)\) gives \(0\) since the coefficients \(1\) and \(-1\) lead to zero.

By combining like terms, you reduce the expression to simpler forms, making it easier to work with or solve. This is a fundamental skill in algebra that helps to clear away complex-looking terms.

Expression Simplification

Simplifying an expression means reducing it to its simplest form where no more combining of like terms is possible. It's like cleaning up algebraic clutter.

• In our solved expression \((9q + p) + (-9q - p)\), after combining like terms, the expression became \(0\). This is the simplest form possible as no terms are left to combine.
• Expression simplification often results in easier number manipulation and problem-solving.

Simplifying expressions is a key skill in algebra that allows mathematicians and students to handle equations efficiently. Proper simplification leads to a clearer understanding of the relationships between terms and prepares the expression for further computation.