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Chapter 1: Problem 5

State the property of real numbers being used. $$(x+2 y)+3 z=x+(2 y+3 z)$$

Short Answer

Expert verified
Associative Property of Addition.

Step by step solution

01

Identify the Property

The given problem requires us to identify a property of real numbers being used in the expression \((x+2y)+3z = x+(2y+3z)\). We need to consider the operations being performed: both sides of the equation resemble a rearrangement without changing the grouped sums of terms. This implies an associative operation.
02

State the Property

The property that describes the rearrangement or regrouping of terms in addition without changing the result is the Associative Property of Addition. This property allows you to change the grouping of numbers being added together and still get the same sum. Formally, the Associative Property of Addition is \((a + b) + c = a + (b + c)\).

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Key Concepts

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

Real Numbers
Real numbers are the set of numbers that include all the numbers we can think of on the number line. This includes:
  • Natural numbers (1, 2, 3, ...)
  • Whole numbers (0, 1, 2, 3, ...)
  • Integers (..., -3, -2, -1, 0, 1, 2, 3, ...)
  • Rational numbers (fractions like \(\frac{1}{2}, \frac{2}{3}\))
  • Irrational numbers (numbers that can't be written as a simple fraction, like \(\sqrt{2}\) or \(\pi\))
Real numbers are important in mathematics as they help us measure and calculate real-world quantities. They can be positive, negative, or zero. The expression given in the problem $$(x+2y)+3z = x+(2y+3z)$$ involves variables that represent real numbers. Understanding real numbers is crucial as they form the foundation for most algebraic operations.
Associative Property of Addition
The Associative Property of Addition is an important concept in algebra. It states that when you add three or more numbers, the way you group them does not affect the sum. In simpler words, you can change the grouping of numbers being added, and it won't change the total.Mathematically, this can be expressed as: \((a + b) + c = a + (b + c)\).In the context of the given expression:
  • The left side: \((x + 2y) + 3z\)
  • The right side: \(x + (2y + 3z)\)
You notice the sum is grouped differently on each side, yet the result remains the same.
This property is very useful when solving equations, as it allows us to rearrange and simplify them more easily. Learning and applying the Associative Property can make complex algebraic tasks much simpler.
Rearrangement of Terms
Rearrangement of terms in an expression means changing the order or grouping of numbers. This is a common technique used in algebra to simplify calculations or solve equations more efficiently.In the expression provided:$$(x+2y)+3z = x+(2y+3z),$$the terms are rearranged to demonstrate the property of grouping.By regrouping the terms:
  • The addition of \(x\), \(2y\), and \(3z\) occurs in different sequences.
  • Using the Associative Property of Addition allows us to rearrange these terms without changing the overall value of the expression.
  • This step is crucial in simplifying expressions and solving equations.
Remember, rearranging terms is not about changing the numbers themselves but about finding a more convenient way to handle them. Mastering this skill will enhance problem-solving abilities in mathematics.

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Most popular questions from this chapter

Show that the equation represents a circle, and find the center and radius of the circle. $$3 x^{2}+3 y^{2}+6 x-y=0$$

Factor the expression completely. $$\left(a^{2}+2 a\right)^{2}-2\left(a^{2}+2 a\right)-3$$

Factor the expression by grouping terms. $$x^{5}+x^{4}+x+1$$

Perform the indicated operations and simplify. $$\left(x^{2}-y+z\right)\left(x^{2}+y-z\right)$$

Factor the trinomial. $$6 y^{2}+11 y-21$$
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