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Chapter 7: Problem 53

In the following exercises, simplify. $$ 9 y+23+(-9 y) $$

Short Answer

Expert verified
23

Step by step solution

01

Identify like terms

Look at the expression and identify the terms that can be combined. Here, the like terms are the terms involving 'y' (i.e., 9y and -9y).
02

Combine like terms

Add the coefficients of the like terms. So, 9y + (-9y) = 0y, which simplifies to 0.
03

Simplify the expression

Now that the terms involving 'y' are combined to zero, the expression simplifies to 0 + 23 which is just 23.

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Key Concepts

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

Identifying Like Terms
When simplifying algebraic expressions, one of the first things we need to do is identify like terms. Like terms are terms that have the same variables raised to the same powers. For example, in the expression \(9y + 23 + (-9y)\), the terms \(9y\) and \(-9y\) are like terms because they both contain the variable \(y\).

Recognizing like terms is crucial because it allows us to combine them, making the expression simpler.

Remember:
  • Like terms must have the exact same variable parts (for example, '3x' and '7x' are like terms, but '3x' and '3y' are not)
  • Constant terms (numbers without variables) are also considered like terms to each other (like 2 and 5)
Knowing how to identify like terms is the first step to making the simplification process straightforward.
Combining Coefficients
The next step in simplifying algebraic expressions is combining coefficients of like terms. Coefficients are the numerical part of the terms. For instance, in \(9y\), the coefficient is 9, and in \(-9y\), the coefficient is -9.

To combine coefficients, simply perform the arithmetic operation indicated between them. In our example, we have: \(9y + (-9y)\). When you add the coefficients (9 and -9), you get 0. This means, \(9y + (-9y) = 0y\), which simplifies to 0.

Key points to remember:
  • Combining means adding or subtracting the coefficients.
  • If the coefficients sum to zero, the term cancels out.
In this case, combining like terms with coefficients yields a simpler expression. It is one less term to deal with.
Algebraic Simplification
After identifying like terms and combining their coefficients, the final step is algebraic simplification. This means rewriting the expression in its simplest form.

Take our example: \(9y + 23 + (-9y)\). We've combined \(9y\) and \(-9y\) to get 0. Now, the expression reduces to just 23.

Simplification ensures the expression is written as clearly and concisely as possible. Here are some additional points to consider:
  • Always perform operations step-by-step, combining like terms first.
  • After combining, look for any further possible simplifications.
  • Remember to reorder terms logically, usually constants (numbers) at the end.
By following these steps, the final simplified version of the original expression \(9y + 23 + (-9y)\) is simply 23.

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