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For the pair and , what is the value of (a₁b₂ − a₂b₁), and what does it tell you?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • A0; the pair has no solution
  • B23; the pair has a unique solution
  • C7; the pair has infinitely many solutions
  • D7; the pair has a unique solution

Step-by-Step Solution

a₁b₂ − a₂b₁ = 3(5) − 4(2) = 15 − 8 = 7. This is the denominator in the cross-multiplication formula; because it is non-zero the lines are neither parallel nor coincident, so there is exactly one solution. Adding instead of subtracting gives the trap value 23; a zero denominator would be the parallel/coincident case.
Answer: D) 7; the pair has a unique solution

Key idea: Cross-multiplication method

For a₁ + b₁ + c₁ = 0 and a₂ + b₂ + c₂ = 0, the solution (when unique) is:

/ (b₁c₂ − b₂c₁) = / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁).

From these:
= (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
= (c₁a₂ − c₂a₁) / (a₁b₂ − a₂b₁)

**Caveat**: this works ONLY if (a₁b₂ − a₂b₁) ≠ 0 (i.e., lines aren't parallel/coincident).

Useful when you've memorised the formula and want to skip elimination steps.

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