Textbook Question
Simplify the algebraic expression. 5(2x - 3) + 7x
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 18
State the name of the property illustrated: (3 • 7) + (4 • 7) = (4 • 7) + (3 •7)
Verified step by step guidance1
Observe the given expression: \( (3 \cdot 7) + (4 \cdot 7) = (4 \cdot 7) + (3 \cdot 7) \). Notice that the two products are being added together on both sides.
Identify that the terms on the left side are \(3 \cdot 7\) and \(4 \cdot 7\), and on the right side, the same terms appear but in reversed order: \(4 \cdot 7\) and \(3 \cdot 7\).
Recall the Commutative Property of Addition, which states that changing the order of addends does not change the sum. In symbolic form, this is \(a + b = b + a\).
Since the expression shows the sum of two products with their order switched but the sum remains the same, this illustrates the Commutative Property of Addition.
Therefore, the property illustrated by the equation is the Commutative Property of Addition.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Commutative Property of Addition
The commutative property of addition states that changing the order of addends does not change the sum. For example, a + b = b + a. In the given expression, the sums on both sides have the same terms added in different orders, illustrating this property.
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Change of Base Property
Distributive Property
The distributive property connects multiplication and addition, stating that a(b + c) = ab + ac. Although the given expression resembles terms from distribution, it primarily shows rearrangement of terms rather than distribution itself.
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Multiplication of Real Numbers
Multiplication of real numbers is commutative, meaning a × b = b × a. This allows factors to be reordered without changing the product, which is essential in understanding why (3 • 7) and (7 • 3) are equivalent in the expression.
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Related Practice
Textbook Question
Evaluate each exponential expression in Exercises 1–22.
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In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x2+8x+15
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Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
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In Exercises 15–58, find each product. (2x−3)(x2−3x+5)
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In Exercises 15–32, multiply or divide as indicated. (x2−4)/(x2−4x+4) ⋅ (2x−4)/(x+2)
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