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Class 10 Pair of Linear Equations in Two Variables: cheat sheet
CBSE Mathematics. Two linear equations in x and y: solve them together, and tell from the coefficients whether there is one solution, none, or infinitely many.
Key ideas
- A pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is two straight lines; a solution is a point on both lines.
- One solution (the lines cross) when a₁/a₂ ≠ b₁/b₂.
- No solution (parallel lines) when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
- Infinitely many solutions (the same line) when a₁/a₂ = b₁/b₂ = c₁/c₂.
- Substitution: make x or y the subject of one equation and put it into the other.
- Elimination: multiply so that x or y has the same coefficient in both, then add or subtract the equations.
- In word problems, first write what x and y stand for, then make two equations from the two facts given.
Remember
| Crossing lines (consistent) | a₁/a₂ ≠ b₁/b₂ |
| Parallel lines (inconsistent) | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
| Same line (dependent) | a₁/a₂ = b₁/b₂ = c₁/c₂ |
Watch out
- Sign slips when subtracting one equation from the other: subtract every term, including the constants.
- Finding x and forgetting to go back for y.
- Not checking the answer in both equations.
Worked example
Solve: x/3 + y/3 = 14 and x/2 − y/2 = −3
Working: multiply the first equation by 3 and the second by 2 to clear the fractions, then eliminate
Answer: x = 18, y = 24
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