One Variable
A one-variable linear equation has the form ax + b = 0, and its only solution is x = −b/a as long as a ≠ 0. The method is mechanical: clear fractions by multiplying through by the LCM of the denominators, expand brackets, gather all variable terms on one side and constants on the other, then divide. The CAT angle is rarely the algebra itself; it is the cleanup. Two edge cases are tested directly. If the variable cancels and you are left with a true statement like 5 = 5, the equation has infinitely many solutions (an identity). If you are left with a false statement like 5 = 7, there is no solution. Keep an eye on sign errors when you move terms across the equals sign and on dividing by a coefficient that could be negative — flipping a sign late is the single most common slip. For speed, isolate the variable mentally rather than writing every line.
✅ 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 |