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Chapter 10: Problem 71

Perform the indicated operations. $$ (5+9 x)+(-4-8 x) $$

Short Answer

Expert verified
The simplified expression is 1 + x.

Step by step solution

01

Group the Like Terms

Identify and group the like terms from the expression. Like terms are the terms that contain the same variable raised to the same power.The given expression is y1. displayed as follows:\[(5 + 9x) + (-4 - 8x)\] Group the constant terms together and the variable terms together:\[5 + (-4) + 9x - 8x\]
02

Perform the Operations for Constants

Now perform the addition and subtraction for the constant terms, which are 5 and -4:\[5 - 4 = 1\]
03

Perform the Operations for Variable Terms

Next, perform the addition and subtraction for the variable terms, which are 9x and -8x:\[9x - 8x = x\]
04

Combine the Results

Combine the results from Step 2 and Step 3 to get the final simplified expression:\[1 + x\]

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Key Concepts

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

Combining Like Terms
Combining like terms is a fundamental concept in algebra. Like terms are terms that contain the same variable raised to the same power. For instance, in the expression (5 + 9x) + (-4 - 8x), the terms 9x and -8x are like terms because they both have the variable x.
On the other hand, 5 and -4 are like terms because they are both constant terms.
When you combine like terms, you simply add or subtract their coefficients. For example, combining 9x and -8x would give you: 9x - 8x = x
This process helps to simplify the expression and makes it easier to work with in further calculations.
Combining like terms is essential for solving equations and simplifying expressions, making it a crucial skill in algebra.
Algebraic Operations
Algebraic operations involve performing arithmetic on algebraic expressions. These operations include addition, subtraction, multiplication, and division.
In the given exercise, we primarily focus on addition and subtraction. For instance, in the expression (5 + 9x) + (-4 - 8x):
  • First, you add the constants 5 and -4: 5 - 4 = 1.
  • Next, you subtract the coefficients of the variable terms: 9x - 8x = x.
By doing these algebraic operations, you simplify the expression to its most basic form.Understanding algebraic operations helps to solve complex problems by breaking them down into simpler steps. It enables you to manipulate and transform expressions systematically.
Distributive Property
The distributive property is another core concept in algebra. It states that a(b + c) is equal to ab + ac. Essentially, you distribute the multiplication over the addition or subtraction inside the parentheses.
In some cases, you will need the distributive property to simplify expressions before combining like terms. For example:
  • If you have 2(3x + 4), you distribute 2 to both terms inside the parentheses: 2 * 3x + 2 * 4, which simplifies to 6x + 8.
While the given exercise (5 + 9x) + (-4 - 8x) did not specifically require the distributive property, understanding this concept is crucial for more complex problems.
The distributive property allows for greater flexibility and efficiency when simplifying expressions and solving equations.

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Most popular questions from this chapter

Write each number as a product of a real number and i. Simplify all radical expressions. $$ \sqrt{-96} $$

Without using a calculator, determine between which two consecutive integers each square root lies. For example, \(\sqrt{75}\) is between 8 and \(9,\) because \(\sqrt{64}=8, \sqrt{81}=9,\) and \(64<75<81\) $$ \sqrt{10.3} $$

Check each equation to see if the given value for \(x\) is a solution. $$ \sqrt{x+2}-\sqrt{9 x-2}=-2 \sqrt{x-1} $$ $$(a) 2 \quad (b) 7$$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. See Example 4 $$ \sqrt[4]{y} \cdot \sqrt[5]{y^{2}} $$

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. See Example 4 $$ 3^{1 / 2} \cdot 3^{3 / 2} $$
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