Questions
5–8 questions per exam paper
Difficulty
Medium
Importance
Core foundation — mandatory topic
Overview
A pair of linear equations in two variables consists of two equations of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Mastery of this topic is essential for algebraic problem solving and serves as the foundation for linear programming and coordinate geometry. The core objective is determining whether systems have unique, infinite, or no solutions based on coefficient ratios.
Graphical Method and Consistency
The graphical method involves plotting two lines on a Cartesian plane to identify their point of intersection. The nature of the solution depends on whether the lines are intersecting, parallel, or coincident.
- Unique solution: a1/a2 ≠ b1/b2 (Intersecting lines)
- No solution: a1/a2 = b1/b2 ≠ c1/c2 (Parallel lines)
- Infinitely many solutions: a1/a2 = b1/b2 = c1/c2 (Coincident lines)
- Consistent system: Has at least one solution
- Inconsistent system: Has no solution
Algebraic Methods: Substitution and Elimination
Substitution and elimination are systematic algebraic techniques to isolate variables when exact intersection coordinates are required. Elimination is generally preferred in competitive exams for its speed and reduced risk of algebraic errors.
- Substitution: Express one variable in terms of the other and substitute into the second equation
- Elimination: Multiply equations by constants to make coefficients of one variable identical for subtraction
- Best for solving linear systems without needing graph paper
- Always cross-check the solution in both original equations
Cross Multiplication Method
This method provides a direct formula to find x and y when the coefficients are known. While powerful, it requires extreme caution with signs, particularly the negative values of constants.
- x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1)
- Ensure equations are in standard form ax + by + c = 0
- Denominator a1b2 - a2b1 cannot be zero for a unique solution
- Memorize the pattern using the 'b c a b' coefficient sequence
Formula Sheet
a1x + b1y + c1 = 0
a1/a2 ≠ b1/b2 (Unique)
a1/a2 = b1/b2 ≠ c1/c2 (Parallel)
a1/a2 = b1/b2 = c1/c2 (Coincident)
x = (b1c2 - b2c1) / (a1b2 - a2b1)
Exam Tip
Always verify your final x and y values by substituting them back into the original equations to guarantee full marks.
Common Mistakes
- Forgetting to rearrange equations into standard form ax + by + c = 0 before applying formulas.
- Sign errors during cross-multiplication, especially when dealing with negative constants.
- Assuming lines are parallel just because the x and y coefficients are proportional without checking the constant term.
More Revision Notes
Ready to test yourself?
Play topic-wise Pair of Linear Equations in Two Variables questions in Aspirant Arcade — gamified MCQ practice.
Download Free