Algebra Fundamentals: Terms, Patterns, and Equations

Adding Like Terms in Algebra

When working with algebraic expressions, like terms are those that have the same pronumeral (letter) and the same powers. To add like terms, you combine their coefficients while keeping the pronumeral and its power unchanged.

For example:

5x + 3x

Because 'x' is the common pronumeral in this expression, you can simply add the coefficients (5 and 3) together. The sum is 8. Since 'x' is present in both terms, it must be included in the final answer. Therefore, the result is:

8x

Subtracting Like Terms in Algebra

Similar to addition, subtracting like terms involves combining terms that share the same pronumeral and powers. You subtract their coefficients while maintaining the pronumeral and its power.

For example:

5x - 3x

Since 'x' is the common pronumeral, you can subtract the coefficients (5 and 3). The difference is 2. Remember to include 'x' in the final answer. Thus, the result is:

2x

Understanding Number Patterns and Sequences

A list of numbers arranged in order according to some rule is called a number pattern or sequence. Each separate number in the pattern or sequence is called a term. To find the terms of a sequence, you must identify and follow the underlying pattern rule.

Example of a Number Pattern

Consider the following sequence:

1, 5, 9, 13, __, __, __

(Hint: The rule here is to add 4 to the previous term. So the next terms would be 17, 21, 25.)

Key Definitions in Algebra

• In algebra, letters are used to represent numbers. These letters are called pronumerals.
• When a pronumeral can have different values, it is also called a variable. (A variable is a pronumeral that can represent any value within a given set.)
• An expression is a combination of numbers and variables connected by the four basic mathematical operations (addition, subtraction, multiplication, and division).
• A term is a combination of numbers and pronumerals that are joined only by multiplication or division.
• Coefficients are the numerical factors in front of the pronumeral in individual terms (e.g., in 5p, 5 is the coefficient).
• A constant term is a term that has no pronumeral (i.e., it's just a number).
• Like terms have the same pronumerals and the same powers.
• An expression involving like terms can be simplified by adding or subtracting the like terms. An expression with no like terms cannot be simplified further.

Example: Identifying Algebraic Components

Consider the expression: 4xy + 3y + 24z + 7

1. State all of the terms. (Answer: 4xy, 3y, 24z, 7)
2. What is the constant term? (Answer: 7)
3. What is the coefficient of x? (Answer: 4)
4. Which pronumeral has the coefficient of 24? (Answer: z)

Introduction to Algebraic Equations

An equation is a mathematical statement that tells us that two things are equal. Equations have a left-hand side (LHS) and a right-hand side (RHS) with an equals sign (=) in the middle. Equations can either be true or false. When solving an equation, the goal is to find the value(s) that make the statement true.