Pair of Linear Equations in Two Variables

Learning Objectives

• Understand the graphical representation of linear equations in two variables and identify conditions for unique, infinite, or no solutions based on line intersections, parallelism, or coincidence.
• Calculate solutions of linear equations using algebraic methods such as substitution and elimination, and determine conditions for consistency and dependency.
• Analyze the coefficients of linear equations to assess the consistency, inconsistency, or dependency of the system using ratios of coefficients.
• Apply linear equations to solve real-life problems, including age-related problems, cost calculations, and geometric problems, by formulating and solving pairs of linear equations.

Chapter Notes

Graphical Solution of Linear Equations

• Graphical Representation: A pair of linear equations in two variables can be represented graphically by two lines on a Cartesian plane.
• Types of Solutions:
  • Unique Solution: The lines intersect at a single point. This indicates a consistent system.
  • No Solution: The lines are parallel and do not intersect. This indicates an inconsistent system.
  • Infinitely Many Solutions: The lines coincide, meaning every point on the line is a solution. This indicates a dependent and consistent system.
• Example:
  • Equations: $x + 3y = 6$ and $2x - 3y = 12$
  • Graphically, these lines intersect at $(6, 0)$, indicating a unique solution.

Algebraic Methods of Solving Linear Equations

• Substitution Method:
  1. Solve one equation for one variable.
  2. Substitute this expression into the other equation.
  3. Solve the resulting equation for the second variable.
  4. Substitute back to find the first variable.
  • Example: Solve $7x - 15y = 2$ and $x + 2y = 3$.
    • From $x + 2y = 3$, $x = 3 - 2y$.
    • Substitute in $7x - 15y = 2$: $7(3 - 2y) - 15y = 2$.
    • Solve for $y$: $y = \frac{19}{29}$.
    • Substitute back: $x = \frac{49}{29}$.
• Elimination Method:
  1. Multiply equations to align coefficients of one variable.
  2. Add or subtract equations to eliminate one variable.
  3. Solve for the remaining variable.
  4. Substitute back to find the other variable.
  • Example: Solve $9x - 4y = 2000$ and $7x - 3y = 2000$.
    • Multiply to align $y$: $27x - 12y = 6000$ and $28x - 12y = 8000$.
    • Subtract to eliminate $y$: $x = 2000$.
    • Substitute back: $y = 4000$.

Conditions for Consistency of Linear Equations

• Coefficient Ratios:
  • Consistent: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
  • Inconsistent: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
  • Dependent: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Application of Linear Equations in Real-Life Problems

• Formulating Problems: Translate real-world situations into linear equations.
• Example Problems:
  • Age Problem: Aftab's age problem can be represented by $S - 7 = 7(t - 7)$ and $S + 3 = 3(t + 3)$.
  • Cost Problem: The cost of pencils and erasers can be represented by $2x + 3y = 9$ and $4x + 6y = 18$.
• Solution: Use algebraic methods to solve these equations and interpret the results in the context of the problem.