Equations and Inequalities Chapter 1

Solving Equations Overview

• Two types of equations presented: linear equations and equations with variables to solve.
• Solutions involve isolating variables and substitution to test compatibility.

Part (i)

• First equation: (3 - t = 3)
  • Rearranged to find (t): (t = 0).
• Second equation: (25 + 3t = 36)
  • Rearrangement leads to (3t = 11) or (t = \frac{11}{3}), indicating potential contradiction.

Part (ii)

• Given linear equations:
  • First: (3x - y = 3)
  • Second: (9x - 3y = 9), which simplifies to (3x - y = 3) showing both equations represent the same line.
• Since both equations are equivalent:
  • Infinite solutions exist; any (x) can yield (y) values based on the relationship (y = 3x - 3).

Part (iii)

• Two equations presented to find (x) and (y):
  • First: (0.2x + 0.3y = 1.3)
    • Multiplying through by 10 gives (2x + 3y = 13) for simplicity.
  • Second: (0.4x + 0.5y = 2.3)
    • Multiplying through by 10 gives (4x + 5y = 23).
• Find (x) from the first equation, expressed as ((13 - 3y)/2).
• Both equations provide relationships to find specific values for (x) and (y).

Simplification Techniques

• Substitute derived values into other equations to check for consistency.
• Rearranging terms helps isolate variables for easier computation.
• Identifying relationships between equations can aid in finding infinite solutions.