Problem 69 - Discrete Mathematics with Applications

Question

Determine if each is a legal expression. $$a+b+c$$

Step-by-Step Solution

Answer

\(a+b+c\) is a legal expression, as it contains valid variables and operations according to the rules of algebra.

Step 1: Identify the components of the expression

The expression is \(a+b+c\). Here, we have three variables: \(a\), \(b\), and \(c\). In addition, we also have two operations: addition.

Step 2: Check the rules of algebra

According to the rules of algebra, the proper use of operators and combining of terms should be followed. In this case, we are adding three terms (\(a\), \(b\), and \(c\)), which is a valid operation in algebra.

Step 3: Conclusion

Since the expression \(a+b+c\) consists of valid variables and operations, it is a legal expression according to the rules of algebra.

Key Concepts

Understanding Variables

In algebraic expressions, variables play a crucial role. A variable is essentially a symbol that stands for a number. It's a place holder that can take on different values. Think of variables as the building blocks of algebra. They allow you to write expressions and equations that can solve real-world problems. In the expression \(a+b+c\), \(a\), \(b\), and \(c\) are variables. These variables represent unknown values and can be any number. Variables are important because:

• They provide flexibility in calculations and models.
• They help generalize mathematical problems.
• They allow for solving complex problems by forming equations.

Understanding variables is key to advancing in algebra and in using mathematics to describe and solve problems.

Exploring Addition in Algebra

Addition in algebra is very similar to addition in arithmetic, only it often involves variables rather than just regular numbers. When you add algebraic expressions, you combine like terms. Like terms mean the terms that have the same variable raised to the same power. For example, in the expression \(a+b+c\), each term is already simplified because they are each distinct variables. Important points about addition in algebra:

• You can rearrange the terms in any order due to the commutative property of addition (i.e., \(a + b = b + a\)).
• Unlike arithmetic, you can only combine like terms. If the terms are unlike, just write them together.
• Algebraic expressions simplify addition by using these properties to deal with variables effectively.

This process makes solving equations a lot easier.

Understanding Algebraic Operations

Algebraic operations are the heart of working with expressions. These operations include addition, subtraction, multiplication, and division. In an expression like \(a+b+c\), addition is the primary operation.Here are some critical aspects about algebraic operations:

• These operations help in manipulating expressions to derive solutions.
• They follow specific rules, such as the order of operations (PEMDAS/BODMAS).
• Each operation has properties, like the associative and commutative properties, that allow flexibility in calculating results.

Understanding and applying these operations are essential for mastering algebra. They help perform calculations efficiently and form the foundation for more complex math concepts.