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A pair of linear equations is INCONSISTENT. Which condition must its coefficients satisfy?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • Aa₁/a₂ = b₁/b₂ ≠ c₁/c₂
  • Ba₁/a₂ ≠ b₁/b₂
  • Ca₁/a₂ = b₁/b₂ = c₁/c₂
  • Da₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂

Step-by-Step Solution

Inconsistent means NO solution, i.e. parallel distinct lines: equal - and -coefficient ratios but a different constant ratio. All three ratios equal would give coincident lines (infinitely many solutions) — swapping 'no solution' with 'infinitely many' is the commonest slip here. a₁/a₂ ≠ b₁/b₂ is the condition for a unique solution.
Answer: A) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Key idea: Elimination method

**Steps**:
1. Multiply one or both equations by chosen numbers so coefficients of ONE variable become equal (in magnitude).
2. Add or subtract to ELIMINATE that variable.
3. Solve the resulting one-variable equation.
4. Plug back to find the other.

**Example**: ; .
- Multiply first by 3, second by 4: .
- Add: → . Substituting back gives .

Elimination is faster than substitution when both equations have non-trivial coefficients.

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