Algebra Lecture Notes

Introduction to Algebra

• Algebra follows arithmetic as the second subject in math.
• It is considered relatively easy by most.
• Variables are fundamental to algebra.

Variables

• A variable represents a numeric value that is unknown.
  • Typically represented by letters (e.g., x).
• Variables can have different values in different contexts/problems.
• Within a single problem, each variable maintains a consistent value.

Notation and Operations

• Avoid using x as a multiplication sign to prevent confusion.
  • Use a dot (.) or asterisk (*) for multiplication of constants.
  • A number next to a variable implies multiplication.*

Order of Operations

• Important for solving algebraic expressions correctly.
• Equal sign (=) indicates that the left side is equivalent to the right side.

Solving Equations

• Equations represent equalities.
• Example: To solve 2x + 4 = 16:
  1. Subtract 4 from both sides: 2x = 12
  2. Divide both sides by 2: x = 6
• Use reverse order of operations for solving.

Inequalities

• Similar to equations but with different result implications:
  • Result indicates that a variable is larger/smaller than a value.
• Special Rule:
  • Multiplying/dividing both sides by a negative flips the inequality sign.

Checking Solutions

• Substitute values back into the original equation to verify solutions.

System of Equations

• Puzzle-like problems involving multiple equations and variables.
• Solutions exist if there are as many unique equations as variables.
• Example Procedure:
  1. Rearrange one equation: x = y + 3
  2. Substitute into another: Replace x in second equation.
  3. Solve for y and then x.
  • Solution: x = 5, y = 2

Conclusion

• Algebra provides tools to solve equations and inequalities.
• Mastery of basic algebraic principles enables you to solve complex problems.