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Chapter 4: Problem 16

For the following problems, simplify each of the algebraic expressions. $$ 9 a+12 a $$

Short Answer

Expert verified
Answer: The simplified expression is \(21a\).

Step by step solution

01

Identify like terms

In the given expression: \(9a+12a\), 9a and 12a are like terms since they have the same variable a.
02

Combine like terms

To simplify the expression, add the coefficients of the like terms (in this case, 9 and 12) together: \(9a + 12a = (9+12)a\)
03

Calculate the sum

Now, compute the actual sum of the coefficients: \((9 + 12) = 21\)
04

Write the simplified expression

Finally, we can write the simplified expression as: \(21a\) So, the simplified expression of \(9a+12a\) is \(21a\).

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Key Concepts

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

Like Terms
When working with algebraic expressions, one essential concept to grasp is "like terms." Simply put, like terms are terms that have the exact same variable raised to the same power. This similarity allows them to be combined through addition or subtraction. For example, in the expression \(9a + 12a\), both terms include the variable \(a\).

Here's a quick way to identify like terms:
  • Look for terms with the identical variable part (e.g., both have \(a\)).
  • Ensure that the powers (exponents) on these variables are the same. In our example, both terms have \(a^1\).
  • Like terms can only be combined with other like terms.
Recognizing like terms is crucial as it sets the stage for simplifying expressions by combining them.
Coefficients
The next important concept when dealing with algebraic expressions is understanding "coefficients." A coefficient is simply a numerical factor that is multiplied by a variable in an algebraic term. In the expression \(9a + 12a\), you can see two coefficients: 9 and 12. These are the numbers in front of the variable \(a\).

Why are coefficients important? Here's a breakdown:
  • Coefficients tell you how many times the variable part is being multiplied.
  • They play a crucial role when combining like terms, as they determine the new coefficient of the combined term.
For solving problems like \(9a + 12a\), the coefficients help us understand how many total units of \(a\) we have when the terms are combined.
Combining Like Terms
Once you've identified the like terms and understood their coefficients, the next step is "combining like terms." This is a fundamental process in simplifying algebraic expressions. It involves adding or subtracting the coefficients of like terms while keeping the common variable part unchanged.

Let's walk through combining like terms with an example:
  • With \(9a + 12a\), we identify them as like terms because they both have \(a\).
  • Add the coefficients 9 and 12: \(9 + 12 = 21\).
  • The result is written with the common variable: \(21a\).
By combining like terms, we've simplified the expression into \(21a\), making it easier to work with or interpret in further calculations.

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Most popular questions from this chapter

For the following problems, simplify each of the algebraic expressions. $$ 1\left(2 x-6 b+6 a^{2} b+8 b^{2}\right)+1\left(5 x+2 b-3 a^{2} b\right) $$

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