Problem 11 - Algebra 1: Concepts and Skills

Question

Prove the theorem. Use the basic axioms of algebra and the definition of subtraction given in Example 1. If \(a, b,\) and \(c\) are real numbers, then \((a-b) c=a c-b c\)

Step-by-Step Solution

Answer

Theorem is proved: \((a - b)c = ac - bc\).

Step 1: Recall the Axioms of Algebra

To start this problem, remember the axioms of algebra and how they are applied. They include the Commutative, Associative, and Distributive laws. From these axioms, we can derive the definition of subtraction that we will use in the next step.

Step 2: Definition of Subtraction

Let's first recall the definition of subtraction. Subtraction is defined as adding the additive inverse. So, \(a-b\) is equivalent to \(a+(-b)\). In this case, \(a-b\) in the equation \((a-b)c\) can be rewritten in the additive inverse form \((a+(-b))c\).

Step 3: Utilize Distributive Property

We will use the Distributive Property of multiplication over addition in this step, which states \(a(b + c) = ab + ac\). In this case, applying to our expression \((a+(-b))c\) we get \(ac+(-b)c\).

Step 4: Redefine using Subtraction

Now we can change the expression back to the traditional subtraction format, so \(ac+(-b)c\) becomes \(ac - bc\), thus proving the theorem: \((a - b)c = ac - bc\).

Key Concepts

Subtraction

When we think of subtraction, it’s often viewed as taking away one quantity from another. However, in algebra, subtraction can be defined more fundamentally as the addition of an additive inverse. For example, the expression \(a - b\) can be rewritten as \(a + (-b)\).
This concept plays a significant role in algebra because it allows us to utilize the properties of addition when working with subtraction, leading to easier manipulation of expressions. By reconceptualizing subtraction this way, we can more smoothly apply addition properties like the commutative and associative properties.
With this foundation, we approach problems by substituting subtraction with its additive inverse form, laying the groundwork for deeper understanding and more advanced algebraic simplification.

Distributive Property

The distributive property is a cornerstone of algebra. It connects multiplication and addition, allowing us to distribute a factor across terms inside parentheses. Mathematically, it’s expressed as \(a(b + c) = ab + ac\).
This property is so useful because it breaks down complex expressions into simpler components. When facing an operation involving multiplication distributed over addition or subtraction, we can rearrange terms to simplify calculations, just like with the expression \((a+(-b))c\). By distribution, this becomes \(ac + (-b)c\).
Mastery of the distributive property enables easier expression expansion and factorization, important in solving equations and understanding polynomial behavior.

Real Numbers

Real numbers form the broad set of numbers that include all the rational and irrational numbers like fractions, integers, and numbers with non-repeating decimals. When working with real numbers, we move beyond the simple integers and into a more comprehensive number system that fully supports operations like addition, subtraction, multiplication, and division (except by zero).
In algebra, the set of real numbers gives us a broad field with infinite possibilities, each number uniquely filling a spot on the number line.

• Rational numbers are those that can be expressed as fractions, including terminating or repeating decimals.
• Irrational numbers cannot be written as simple fractions and often have non-repeating, non-terminating decimals (like \(\pi\) and \(\sqrt{2}\)).

Understanding real numbers is crucial for algebra as operations must be able to handle this full spectrum of numbers, ensuring that we correctly apply rules, like the distributive property, across them.