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Chapter 1: Problem 70

Explain how the distributive property is used to combine like terms. Give an example.

Short Answer

Expert verified
Using the distributive property, combine like terms 3x and 5x to get 8x.

Step by step solution

01

- Show the Expression

Start with an algebraic expression that can be simplified by combining like terms, for example: \[3x + 5x\]
02

- Identify Like Terms

Recognize that 'like terms' are terms that have the same variable raised to the same power. In this case, both terms are like terms because they both have the variable \(x\).
03

- Factor out the Common Variable

Use the distributive property to factor out the common variable. The expression can be rewritten as:\[3x + 5x = x(3 + 5)\]
04

- Simplify Inside the Parentheses

Add the coefficients inside the parentheses:\[x(3 + 5) = x \times 8\]
05

- Write the Final Simplified Expression

Multiply the variable by the simplified coefficient:\[x \times 8 = 8x\]

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Key Concepts

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

Combining Like Terms
When working with algebraic expressions, it's crucial to combine like terms to simplify them. Like terms have the same variable raised to the same power. For instance, in the expression \(3x + 5x\), both terms have the variable \(x\), making them like terms. Simplifying like terms helps in solving equations efficiently. To combine like terms, add their coefficients together. First, identify the like terms. Next, add their numerical coefficients while keeping the variable part the same. For example, for the expression \(3x + 5x\), add the coefficients 3 and 5, resulting in \(8x\). This helps to ease the calculation process in further steps.
Factoring
Factoring in algebra involves breaking down an expression into simpler components called factors. It's like reversing multiplication to find what was multiplied to get the original expression. In the context of combining like terms, using the distributive property is a common method. The distributive property states that \(a(b + c) = ab + ac\). To apply this, find the common variable or factor within an expression. For example, in \(3x + 5x\), factor out the common variable \(x\). This gives: \(x \times (3 + 5)\). Factoring makes it easier to work with and solve expressions, especially when dealing with complex equations.
Algebraic Expression
An algebraic expression is a combination of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. For instance, \(3x + 5\) is an algebraic expression where \(3\) and \(5\) are constants, and \(x\) is the variable. Expressions can be simplified by combining like terms and using operations such as factoring. Simplifying expressions provides a clearer and more straightforward form that is easier to solve. Familiarity with terms, coefficients, and variables is essential. A well-simplified algebraic expression aids in understanding and efficiently tackling more advanced algebraic problems.

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Most popular questions from this chapter

Graph the elements of each set on a number line. $$ \left\\{-\frac{2}{3}, 0, \frac{4}{5}, \frac{12}{5}, \frac{9}{2}, 4.8\right\\} $$

Graph the elements of each set on a number line. $$ \left\\{-\frac{6}{5},-\frac{1}{4}, 0, \frac{5}{6}, \frac{13}{4}, 5.2, \frac{11}{2}\right\\} $$

Graph the elements of each set on a number line. $$ \\{-4,-2,0,3,5\\} $$

Exercises \(91-116\) provide more practice on operations with fractions and decimals. Perform the indicated operations. $$(-4.2)(1.4)(2.7)$$

An approximation of the average price of a theater ticket in the United States from 1977 through 2007 can be obtained by using the expression. $$ 0.1399 x-274.4 $$ where \(x\) represents the year. (Source: National Association of Theater Owners.) (a) Use the expression to complete the table. Round answers to the nearest cent. (b) How has the average price of a theater ticket in the United States changed from 1977 to \(2007 ?\)
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