Simultaneous Equations
A pair of linear equations represents two straight lines, and the number of solutions is just how those lines sit. Compare the coefficient ratios a₁/a₂, b₁/b₂ and c₁/c₂. If a₁/a₂ ≠ b₁/b₂ the lines cross once, giving exactly one (unique) solution — the consistent and independent case. If a₁/a₂ = b₁/b₂ but this differs from c₁/c₂, the lines are parallel and never meet, so there is no solution (inconsistent). If all three ratios are equal, the equations are really the same line written twice, so there are infinitely many solutions (dependent). A quick equivalent for the unique case is the determinant D = a₁b₂ − a₂b₁ ≠ 0. CAT loves the "find k" version: it gives a parameter inside the coefficients and asks for the value that makes the system have no solution or infinitely many. The drill is to set the matching ratios equal, solve for k, then check the c-ratio to decide which case you have landed in.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Single and standard forms
| Linear equation (one variable) | ax + b = 0 ⇒ x = −b/a (a ≠ 0) |
|---|---|
| Two-variable standard form | a₁x + b₁y = c₁ and a₂x + b₂y = c₂ |
| Slope–intercept line | y = mx + c (m = slope, c = y-intercept) |
| Cross-multiplication solution | x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) |
| Companion value | y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁) |
Solution conditions for a pair
| Unique solution (lines meet) | a₁/a₂ ≠ b₁/b₂ |
|---|---|
| No solution (parallel lines) | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
| Infinite solutions (same line) | a₁/a₂ = b₁/b₂ = c₁/c₂ |
| Determinant test | D = a₁b₂ − a₂b₁; D ≠ 0 ⇒ unique |
| n equations need | n independent equations for n unknowns |