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Chapter 4: Problem 44

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

Short Answer

Expert verified
The expression simplifies to 0.

Step by step solution

01

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.
02

Group Like Terms

Group together the terms containing \(q\) and the terms containing \(p\). The expression now looks like this: \((9q) + (-9q) + (p) + (-p)\).
03

Combine Like Terms

Simplify each group of like terms separately. The terms \((9q) + (-9q)\) combine to 0, and \((p) + (-p)\) also combine to 0.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals.$$ \left\\{\begin{array}{l} x-\frac{y}{3}=-1 \\ -\frac{x}{2}+\frac{y}{8}=\frac{1}{4} \end{array}\right. $$

Write equivalent equations by multiplying both sides of each given equation by the given nonzero number. $$ 5 a-7 b=-4 \text { by }-4 $$

As you solve the system \(\left\\{\begin{array}{c}3 x-y=-6 \\ -3 x+2 y=7\end{array},\right.\) you find that \(y=1\). Is this the solution to the system?

Rewrite each sentence using mathematical symbols. Do not solve the equations. If the quotient of twice a number and 7 is subtracted from the reciprocal of the number, the result is 2 .

Rewrite each sentence using mathematical symbols. Do not solve the equations. Twice a number, added to 6 , is 3 less than the number.
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