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Chapter 1: Problem 88

Which property of real numbers justifies each statement? $$x(y+z)=(y+z) x$$

Short Answer

Expert verified
The property of real numbers that justifies the equation \(x(y+z)=(y+z)x\) is the Commutative Property of Multiplication.

Step by step solution

01

Identify the Structure of the Equation

Notice the structure of the equation: \(x(y+z)=(y+z) x\). Despite the different order of multiplication, both sides of the equation result in the same value. The main change is the order of the operation - specifically, the placement of the variable \(x\).
02

Note the Commutative Property of Multiplication

In the real number system, the commutative property of multiplication indicates that the order of multiplication does not affect the product. That is, for any real numbers \(a\) and \(b\), \(a \cdot b = b \cdot a\). This property seems to be the one justifying the given equation.
03

Apply the Commutative Property to the Equation

Applying the commutative property of multiplication to the given equation \(x(y+z)=(y+z)x\), we can see that it justifies this equality. By this property, the variable \(x\) can be on either the left or the right of the other terms without altering the result of the equation, which corresponds to the given equation.

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Key Concepts

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

Commutative Property
The commutative property is a fundamental aspect of mathematics, especially when dealing with real numbers. This property states that the order in which two numbers are multiplied does not affect the product. An easy way to understand this is through a simple example:
  • If you multiply 3 and 4, whether you arrange it as 3 脳 4 or 4 脳 3, the result is always 12.

This property holds true for any two real numbers, not just whole numbers or integers. It's important because it allows flexibility in how you solve equations or problems. For instance, if you're rearranging terms:
  • This helps simplify expressions, especially when combined with other properties like associativity or distributiveness.
  • For example, if you have an equation like \(x(y+z)=(y+z)x\), you can switch the position of the \(x\) without changing the result due to the commutative property.
Multiplication
Multiplication is one of the basic arithmetic operations that you learn early in math. In essence, it's repeated addition. For example, 4 multiplied by 3 can be interpreted as adding 4 three times:
  • That is, 4 + 4 + 4, which equals 12.

In algebra, multiplication takes on a powerful role because of its interaction with properties like commutativity, associativity, and distributivity. It allows you to simplify complex equations by rearranging and grouping terms. This makes solving algebra equations more manageable.
  • Understanding how multiplication behaves, especially with variables and constants, is essential for solving equations.
  • For example, using the commutative property of multiplication, you can rewrite and simplify expressions such as \(x(y+z)\) to \((y+z)x\) without affecting the result.
Algebra Equations
Algebra equations are mathematical statements that express the equality between two expressions. These equations often involve variables, which represent unknown values that we're trying to solve for. The power of algebra is its ability to describe relationships and patterns using these variables.
  • For example, in the equation \(x + 3 = 7\), algebra helps us find that \(x\) must be 4 to make the statement true.

In the context of properties like commutative property, algebra equations showcase how relationships remain unchanged despite rearranging elements around the equal sign. This is particularly useful:
  • It allows us to manipulate expressions and solve equations efficiently.
  • For instance, in the equation \(x(y+z) = (y+z)x\), we see how the commutative property enables reordering without altering the outcome. This flexibility is vital for solving complex algebraic expressions.

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Most popular questions from this chapter

Consider the following sets: the integers, natural numbers, even and odd integers, positive and negative numbers, prime and composite numbers, and rational numbers. Find a number that fits in as few of these categories as possible.

A rope \(x\) feet long is cut into 5 equal pieces. Find an expression for the length of each piece.

Write each inequality as an equivalent inequality in which the inequality symbol points in the opposite direction. $$6>0$$

Write each inequality as an equivalent inequality in which the inequality symbol points in the opposite direction. $$\frac{12}{4}<\frac{24}{6}$$

Graph each set of numbers on a number line. Use brackets or parentheses where applicable. The real numbers greater than or equal to -5 and less than 4
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