Linear Equations in One Variable

• Basic Concepts

  • Algebraic Expression: Combination of variables and constants (e.g., $2x + 3$).
  • Equation: Equality statement containing an '=' sign (e.g., $2x + 3 = 7$).
  • Linear Equation: Equation where highest degree (power) of variable is 1.
  • One Variable: Contains only one unknown letter (e.g., $x$, $y$, or $z$).
  • Anatomy of Equation
    • LHS: Left Hand Side (expression on the left of '=').
    • RHS: Right Hand Side (expression on the right of '=').
    • Equality Sign: Indicates LHS value equals RHS value.

• Methods of Solving

  • Variables on One Side
    • Keep variable term on LHS and numbers on RHS.
    • Transposition Method: Move terms across '=' by reversing operation:
      • Addition ($+$) $\rightarrow$ Subtraction ($-$)
      • Subtraction ($-$) $\rightarrow$ Addition ($+$)
      • Multiplication ($\times$) $\rightarrow$ Division ($\div$)
      • Division ($\div$) $\rightarrow$ Multiplication ($\times$)
  • Variables on Both Sides
    • Transpose all variable terms to LHS.
    • Transpose all constant terms to RHS.
    • Combine like terms and solve for the variable.
  • Equations Reducible to Linear Form
    • Applied to rational/fractional expressions: $\frac{ax + b}{cx + d} = \frac{p}{q}$
    • Use Cross-Multiplication: $q(ax + b) = p(cx + d)$

• Applications (Word Problems)

  • Steps to Solve
    1. Read problem carefully and identify unknown.
    2. Assume the unknown as a variable 'x'.
    3. Form an equation based on problem condition.
    4. Solve the equation for 'x'.
    5. Verify the answer against original condition.
  • Types of Word Problems
    • Number Systems: Consecutive numbers, sum/difference of digits.
    • Age Problems: Present age vs. future/past age ratios.
    • Geometry: Perimeter, side lengths of shapes.
    • Currency & Money: Denomination counts and total value.
    • Speed & Distance: Time, speed, and stream problems.

• Key Properties

  • Solution / Root: Value of variable that makes $\text{LHS} = \text{RHS}$.
  • Balance Principle: Equation remains unchanged if same operation is performed on both sides:
    • Add/subtract same number on both sides.
    • Multiply/divide both sides by same non-zero number.