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Chapter 1: Problem 74

Simplify each expression. $$ 6(3 p-2)-(5 p+1) $$

Short Answer

Expert verified
13p - 13

Step by step solution

01

Distribute the 6

Multiply each term inside the parenthesis by 6: 6(3p - 2) = 18p - 12
02

Rewrite the expression

Substitute the result from Step 1 back into the original expression: 18p - 12 - (5p + 1)
03

Distribute the negative sign

Distribute the negative sign inside the parenthesis: 18p - 12 - 5p - 1
04

Combine like terms

Combine the terms involving p and the constant terms: (18p - 5p) + (-12 - 1) = 13p - 13

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Key Concepts

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

Simplification
Simplification is the process of reducing an algebraic expression to its simplest form. This often involves removing parentheses, combining like terms, and performing basic arithmetic. In the given exercise, we have the expression: \[ 6(3p-2)-(5p+1) \]. Simplifying this involves distributing and combining like terms systematically. Simplification helps to make expressions more manageable and straightforward for further calculations.
Distribution
Distribution is a key algebraic principle that helps in expanding expressions. It involves multiplying each term inside a parenthesis by a factor outside the parenthesis. For example, in the exercise, we start with: \[ 6(3p-2)-(5p+1) \]. We distribute the 6 across the terms inside the parenthesis: \[ 6 \times 3p \] and \[ 6 \times -2 \], to get: \[ 18p - 12 \]. Distribution is like dealing out cards; each term inside the parenthesis gets its share of the multiplier.
Combining Like Terms
Combining like terms is another crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For instance, \[ 18p \] and \[ -5p \] are like terms because they both contain the variable \[ p \]. We combine them by adding or subtracting their coefficients. In our exercise, after we distribute and rewrite the expression, \[ 18p - 12 - 5p - 1 \], we combine like terms: \[ (18p - 5p) + (-12 - 1) \] to get: \[ 13p - 13 \]. This step makes the expression simpler and easier to work with.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 3 if the sum of its digits is divisible by \(3,\) and not otherwise. Show that (a) \(4,799,232\) is divisible by 3 and (b) \(2,443,871\) is not divisible by 3

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