Pair of Linear Equations in Two Variables — Notes
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3.1 Pair of Linear Equations
A pair of linear equations in x and y can be written as a1x+b1y+c1=0 and a2x+b2y+c2=0. A solution is an ordered pair satisfying both equations.
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3.2 Graphical Interpretation
Each linear equation represents a straight line. Intersecting lines have one solution; parallel distinct lines have no solution; coincident lines have infinitely many solutions.
Consistency conditions
Unique solution: a1/a2 ≠ b1/b2.
No solution: a1/a2 = b1/b2 ≠ c1/c2.
Infinitely many: all three ratios are equal.
No solution: a1/a2 = b1/b2 ≠ c1/c2.
Infinitely many: all three ratios are equal.
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3.3 Substitution Method
Solve one equation for one variable and substitute it into the other equation. This is especially convenient when one coefficient is 1 or −1.
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3.4 Elimination Method
Multiply one or both equations so that the coefficients of one variable become equal or opposite. Add or subtract to eliminate that variable, then substitute back.
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3.5 Cross-Multiplication Method
For a1x+b1y+c1=0 and a2x+b2y+c2=0, cross multiplication can be used carefully to obtain x and y when a unique solution exists.
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3.6 Word Problems
Translate statements into equations before solving. Typical applications involve ages, numbers, costs, distances, and quantities. Define variables clearly and check the final values in the original conditions.
⚡ Quick Revision
- Two equations represent two lines.
- Intersecting → one solution; parallel → no solution; coincident → infinitely many.
- Use substitution, elimination or cross multiplication.
- Always verify a word-problem answer in the original statements.
🎯 Board Exam Focus
- Check the ratio conditions for consistency questions.
- Keep signs organised while eliminating variables.
- Draw a rough graph when the nature of solutions is asked.
🚀 Last-Minute Revision
✔ Read every formula with its conditions and symbols.
✔ Write given data first and keep units consistent.
✔ For proofs, state the theorem and show logical steps.
✔ Check signs, substitutions and final units.
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