Pair of Linear Equations in Two Variables

• General Form

  • Equation 1: $a_1x + b_1y + c_1 = 0$
  • Equation 2: $a_2x + b_2y + c_2 = 0$
  • Variables: $x, y$
  • Constants: $a, b, c$ are real numbers
  • Condition: $a^2 + b^2 \neq 0$

• Nature of Solutions (Ratio Comparison)

  • Unique Solution (Intersecting Lines)
    • Condition: $a_1/a_2 \neq b_1/b_2$
    • System: Consistent
  • Infinitely Many Solutions (Coincident Lines)
    • Condition: $a_1/a_2 = b_1/b_2 = c_1/c_2$
    • System: Dependent & Consistent
  • No Solution (Parallel Lines)
    • Condition: $a_1/a_2 = b_1/b_2 \neq c_1/c_2$
    • System: Inconsistent

• Methods of Solving

  • Graphical Method
    • Plotting both lines on a Cartesian plane
    • Point of intersection is the solution $(x, y)$
  • Algebraic Methods
    • Substitution Method: Express one variable in terms of the other and substitute
    • Elimination Method: Multiply equations to make coefficients of one variable equal, then add/subtract
    • Cross-Multiplication Method: Using the formula based on coefficients (Note: Check latest MP Board syllabus for deletions)

• Equations Reducible to Linear Form

  • Used for variables in the denominator (e.g., $1/x, 1/y$)
  • Substitution: Let $1/x = u$ and $1/y = v$
  • Solve for $u$ and $v$, then find $x$ and $y$

• Word Problem Applications

  • Age Problems: Relations between past, present, and future ages
  • Number/Digit Problems: Reversing digits ($10y + x$)
  • Fraction Problems: Changes in numerator and denominator
  • Speed, Distance, Time: Upstream and Downstream boat problems
  • Fixed & Variable Charges: Taxi fares, library fees, or hostel charges
  • Geometry Based: Perimeter and area relations