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Chapter 1: Problem 39

Determine which of the fundamental laws of algebra is demonstrated. $$6(3+1)=6(3)+6(1)$$

Short Answer

Expert verified
The fundamental law of algebra demonstrated is the Distributive Property.

Step by step solution

01

Recognizing the Structure of the Equation

The equation given is \(6(3+1) = 6(3) + 6(1)\). We need to recognize this as a common algebraic property. Look for a pattern that could correspond to one of the fundamental laws of algebra.
02

Identify the Algebraic Property

Notice that the structure: \(a(b+c) = a \, b + a \, c\), where in this example \(a=6\), \(b=3\), and \(c=1\), is indicative of a fundamental algebraic property known as the Distributive Property.
03

Confirm Using the Distributive Property

The Distributive Property states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the results. So, \(6(3+1) = 6 \times 3 + 6 \times 1\) confirms this property is used.

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Key Concepts

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

Algebraic Properties
In algebra, properties help simplify expressions and solve equations. They provide a set of rules that can be applied across various mathematical problems. Understanding these properties is essential, as they form the backbone of algebraic manipulations. One such fundamental property is the Distributive Property, which is often used to expand algebraic expressions or simplify them.

The Distributive Property is expressed as follows:
  • If you have a number multiplying a sum, such as \(a(b+c)\), you can distribute the multiplication across the addition: \(a \cdot b + a \cdot c\).
  • This is extremely useful in simplifying expressions and making calculations more manageable.
Consider how the above property can simplify complex expressions. It's about breaking down the problem into smaller, easy-to-manage pieces using consistent rules.
Fundamental Laws of Algebra
The fundamental laws of algebra include a set of core principles that underpin all algebraic operations. These are:
  • Commutative Law: The order of addition or multiplication does not affect the result. For addition, \(a+b = b+a\). For multiplication, \(a \cdot b = b \cdot a\).
  • Associative Law: The way numbers are grouped does not affect their sum or product. For addition, \((a+b)+c = a+(b+c)\). For multiplication, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
  • Distributive Law: As mentioned earlier, this law allows us to multiply a sum by distributing the multiplication: \(a(b+c) = a \cdot b + a \cdot c\).
These laws guarantee that no matter how complex the problem, the operations will yield consistent results. They are like the grammar rules for the language of algebra, ensuring clarity and order.
Mathematical Proofs
In mathematics, a proof is a logical argument that verifies the truth of a statement. Mathematical proofs utilize a series of logical steps to ensure that the conclusion follows inevitably from the premises. When we talk about proving things such as the Distributive Property, we are essentially demonstrating why the property holds true in all cases.

For example, the proof of the Distributive Property relies on basic arithmetic and logical operations:
  • You begin by assuming three numbers \(a\), \(b\), and \(c\).
  • By definition, \(a(b+c)\) is the same as \(a \cdot b + a \cdot c\).
  • We use your existing knowledge of arithmetic to show that both sides produce the same result regardless of the numbers chosen.
Using proofs in mathematics fortifies the concepts and ensures we aren't merely relying on pattern recognition, but true understanding. This makes mathematical proofs a crucial part of learning algebra.

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Most popular questions from this chapter

Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. A car's radiator contains 12 L of antifreeze at a \(25 \%\) concentration. How many liters must be drained and then replaced by pure antifreeze to bring the concentration to \(50 \%\) (the manufacturer's "safe" level)?

Simplify the given algebraic expressions. For the following expressions, subtract the third from the sum of the first two: \(3 a^{2}+b-c^{3}, 2 c^{3}-2 b-a^{2}, 4 c^{3}-4 b+3\).

Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. Before being put out of service, the supersonic jet Concorde made a trip averaging \(120 \mathrm{mi} / \mathrm{h}\) less than the speed of sound for \(0.1 \mathrm{h}\) and \(410 \mathrm{mi} / \mathrm{h}\) more than the speed of sound for \(3.0 \mathrm{h}\). If the trip covered 3990 mi, what is the speed of sound?

Answer the given questions. Is the following simplification correct? Explain. $$\begin{aligned} 2 a-3 b-4 c-(-5 a+3 b-2 c) &=2 a-3 b-4 c+5 a-3 b-2 c \\ &=7 a-6 b-6 c \end{aligned}$$

Solve the given problems. A car costs \(\$ 38,000\) new and is worth \(\$ 24,000\) 2 years later. The annual rate of depreciation is found by evaluating \(100(1-\sqrt{V / C}),\) where \(C\) is the cost and \(V\) is the value after 2 years. At what rate did the car depreciate? (100 and 1 are exact.)
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