Problem 22 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$5 \star+2 \Delta+3 \Delta-8 \star$$

Step-by-Step Solution

Answer

The simplified expression is \(-3 \star + 5 \Delta\).

Step 1: Identify Like Terms

The expression given is \(5 \star + 2 \Delta + 3 \Delta - 8 \star\). Look for terms that have the same symbol or variable accompanying them. Here, \(\star\) and \(\Delta\) are the terms to be combined separately.

Step 2: Combine Like Terms with \(\star\)

Identify the terms with \(\star\). These are \(5 \star\) and \(-8 \star\). Combine them by performing the operation: \(5 - 8 = -3\). The result is \(-3 \star\).

Step 3: Combine Like Terms with \(\Delta\)

Identify the terms with \(\Delta\). These are \(2 \Delta\) and \(3 \Delta\). Combine them by performing the addition: \(2 + 3 = 5\). The result is \(5 \Delta\).

Step 4: Write the Simplified Expression

Combine the results from combining like terms: \(-3 \star\) and \(5 \Delta\). The simplified expression is \(-3 \star + 5 \Delta\).

Key Concepts

Combining Like Terms

To simplify an algebraic expression, one of the first steps is to combine like terms. But what exactly does this mean? Like terms are terms that have the same variables, raised to the same power. In other words, you can only combine terms that are alike in these ways. The coefficients (the numbers in front of the variables) do not need to match.
So, when you're looking at an expression like \(5 \star + 2 \Delta + 3 \Delta - 8 \star\), identify and group the like terms. Here, the terms \(5 \star\) and \(-8 \star\) are like terms because they both have the same variable, \(\star\). Similarly, \(2 \Delta\) and \(3 \Delta\) are like terms because they both involve \(\Delta\).

• Terms with the same variables are like terms.
• Combine them by adding or subtracting their coefficients.
• Keep the variable part the same.

Once identified, perform the arithmetic with their coefficients to combine them, simplifying the expression.

Algebraic Expressions

Algebraic expressions are a fundamental component of algebra. They consist of numbers, variables (like \(x\), \(y\), or symbols \(\star\), \(\Delta\) in this case), and operations (like addition, subtraction, multiplication, and division). An expression is different from an equation as it does not include an equality sign. Instead, it represents a mathematical phrase that communicates a particular quantity or operation.
For instance, in the expression \(5 \star + 2 \Delta + 3 \Delta - 8 \star\), "\(5 \star\)" and "\(2 \Delta\)" are separate units that, when combined, simplify to another expression.

• An expression does not have an equal sign.
• It is a statement of value using numbers and variables.
• Mathematical operations tie the terms together.

Mastering algebraic expressions involves understanding how to manipulate and simplify them through arithmetic processes such as adding or subtracting like terms.

Basic Algebra

Basic algebra covers fundamental operations involving variables and numbers. It serves as the foundation for nearly all higher-level mathematics. The basic idea is to solve problems and understand mathematical relationships by representing numbers through symbols or variables.
Simplifying algebraic expressions, like in our exercise, is a crucial skill in basic algebra. This exercise involves recognizing patterns and applying simple operations to clarify or shorten expressions.

• Learn to identify like terms.
• Understand operations with variables.
• Practice combining powers, coefficients, and terms.

Once you get hold of the basic principles of algebra, you can tackle more complex equations and problems with confidence. Understanding these concepts simplifies mathematical ideas, providing the tools necessary for solving equations and understanding functions.