Problem 84
Question
Name the property of real numbers illustrated by each equation. $$ 3(2 x+y)=6 x+3 y $$
Step-by-Step Solution
Verified Answer
The property of real numbers illustrated by the equation is the Distributive Property.
1Step 1: Identify the Mathematical Operation
The given equation is using the multiplication operation where a single term is being distributed across terms inside a parenthesis. This is known as the distributive property.
2Step 2: Recognize the Real Number Property
Since multiplication is being distributed over addition in the expression, the real number property illustrated here is the Distributive Property of Multiplication over Addition.
Key Concepts
Real Numbers PropertiesAlgebraic ExpressionsMathematical Operations
Real Numbers Properties
When it comes to mathematics, understanding the underlying properties of real numbers is essential in solving algebraic equations efficiently.
Real numbers include the whole numbers, fractions, irrational numbers, and decimals that we encounter daily in mathematics. Fundamental to working with these numbers are properties such as the commutative, associative, distributive, identity, and inverse properties. In the given exercise, we focus on the distributive property, which allows us to multiply a number by a group of numbers added together by distributing the multiplication over each addend separately.
This property is expressed in the general form as follows: \( a(b + c) = ab + ac \). It paves the way to simplify expressions and solve equations involving both multiplication and addition. By applying this property, students can break down more complex problems into simpler steps that are more manageable.
Real numbers include the whole numbers, fractions, irrational numbers, and decimals that we encounter daily in mathematics. Fundamental to working with these numbers are properties such as the commutative, associative, distributive, identity, and inverse properties. In the given exercise, we focus on the distributive property, which allows us to multiply a number by a group of numbers added together by distributing the multiplication over each addend separately.
This property is expressed in the general form as follows: \( a(b + c) = ab + ac \). It paves the way to simplify expressions and solve equations involving both multiplication and addition. By applying this property, students can break down more complex problems into simpler steps that are more manageable.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Variables represent unknown values and can take on any value of the numbers they represent. Operations include addition, subtraction, multiplication, and division.
Understanding how to work with algebraic expressions is pivotal for solving equations and inequalities. In our example, \( 3(2x + y) \) is an algebraic expression where \( 3 \) is a coefficient being multiplied with the terms inside the parentheses. Emphasizing the structure and components of these expressions helps students recognize patterns and apply the correct operations to simplify or solve them. Breakdown of such expressions using the distributive property elucidates how complex expressions are just a combination of simpler ones connected through basic operations.
Understanding how to work with algebraic expressions is pivotal for solving equations and inequalities. In our example, \( 3(2x + y) \) is an algebraic expression where \( 3 \) is a coefficient being multiplied with the terms inside the parentheses. Emphasizing the structure and components of these expressions helps students recognize patterns and apply the correct operations to simplify or solve them. Breakdown of such expressions using the distributive property elucidates how complex expressions are just a combination of simpler ones connected through basic operations.
Mathematical Operations
Mathematical operations are the building blocks for all of math. They include addition, subtraction, multiplication, and division. With regards to the distributive property, it specifically deals with two operations: multiplication and addition.
Understanding how to move between these operations is crucial. When students master the distributive property, they gain the ability to simplify algebraic expressions and solve equations more efficiently. This property is particularly useful when dealing with algebraic expressions that involve both addition and multiplication, as seen in the exercise. By distributing the multiplication over the addition, \( 3(2x + y) = 6x + 3y \), students convert a single, more complex expression into a simpler one, leading to a deeper understanding and quicker solutions.
Understanding how to move between these operations is crucial. When students master the distributive property, they gain the ability to simplify algebraic expressions and solve equations more efficiently. This property is particularly useful when dealing with algebraic expressions that involve both addition and multiplication, as seen in the exercise. By distributing the multiplication over the addition, \( 3(2x + y) = 6x + 3y \), students convert a single, more complex expression into a simpler one, leading to a deeper understanding and quicker solutions.
Other exercises in this chapter
Problem 84
Each matrix represents vertices of a polygon. Translate each figure 3 units right and 2 units down. Express your answer as a matrix. $$ \left[\begin{array}{rrr}
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Graph each function. $$ y=-2(x+1)^{2}-3 $$
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Graph each function. $$ y=\frac{1}{2}(x-4)^{2}+1 $$
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Name the property of real numbers illustrated by each equation. $$ 3 x^{2}+7 y=7 y+3 x^{2} $$
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