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Chapter 5: Problem 79

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ (9 a+6 b-5)+(-11 a-7 b+6) $$

Short Answer

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

Step by step solution

01

Identify like terms

To solve the expression \((9a + 6b - 5) + (-11a - 7b + 6)\), first identify terms that are similar. That is, look for terms with the same variable raised to the same power. In this expression, the like terms are \(9a\) and \(-11a\), \(6b\) and \(-7b\), and the constants \(-5\) and \(6\).
02

Combine like terms

Now, add or subtract the coefficients of the like terms:- For the \(a\) terms: \(9a + (-11a) = -2a\).- For the \(b\) terms: \(6b + (-7b) = -1b\). This simplifies to \(-b\).- For the constants: \(-5 + 6 = 1\).
03

Write the simplified expression

Combine the results from Step 2 to form the simplified expression:\(-2a - b + 1\).

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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, a vital skill is identifying "like terms." Like terms are elements of the expression that have the same variable raised to the identical power. This uniformity allows them to be grouped and simplified easily.
For instance, in the expression \((9a + 6b - 5) + (-11a - 7b + 6)\), you can spot the like terms by looking at the variables:
  • \(9a\) and \(-11a\) are like terms because they both have the variable \(a\) raised to the first power.
  • \(6b\) and \(-7b\) are another set of like terms, both sharing the variable \(b\).
  • The numbers \(-5\) and \(6\) are also like terms since they are constant terms or numbers without a variable.
Recognizing like terms is the first and crucial step to simplifying complex expressions effectively. Without this, you cannot proceed to solve the exercise correctly.
Combining Like Terms
Once you have identified the like terms, the next step is to "combine like terms." This involves adding or subtracting the numerical coefficients of these terms.
By combining like terms, you simplify the expression and reduce the complexity of the problem. Let's take a look at how this works in our given expression:
  • Combine the \(a\) terms: \(9a + (-11a)\). Keep the variable "a" constant and perform the arithmetic on the coefficients: \(9 - 11 = -2\), thus resulting in \(-2a\).
  • For the \(b\) terms, \(6b + (-7b)\): Perform \(6 - 7 = -1\), which simplifies to \(-b\).
  • Finally, combine the constant numbers \(-5 + 6\): Simply calculate \(-5 + 6 = 1\).
Combining like terms reduces the expression to its simplest form, making further calculations, if needed, much easier. It is a basic yet powerful operation in algebra.
Simplifying Expressions
The ultimate goal of dealing with algebraic expressions is often to "simplify expressions." This means reducing the expression to its simplest form where no further operations on like terms can be performed.
After you have combined like terms, you will write down the simplified version of the expression. In the original exercise:
  • The expression \((9a + 6b - 5) + (-11a - 7b + 6)\) simplifies to \(-2a - b + 1\) after combining like terms.
Here, each component \(-2a\), \(-b\), and \(1\) stands alone where no further combining is possible, demonstrating that the expression is now in its simplest form.
By simplifying, you've achieved clarity and concise representation of the algebraic expression. Mastering this will streamline solving equations, graphing, and more advanced algebraic operations.

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Most popular questions from this chapter

Multiply. $$(2 x-5)^{3}$$

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \frac{1.4 \times 10^{-2}}{7 \times 10^{-8}} $$

Perform the indicated operation. $$(5 x)^{2}$$

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ (3 x-2+6 y)+(7 x-2-y) $$

Simplify each expression. See Section \(5.1 .\) $$ \frac{x^{3} y^{6}}{x y^{2}} $$
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