Linear Equations
Linear equations are the workhorse of CAT Algebra. A linear equation is one in which every variable appears only to the first power — no squares, no products of two variables, no variables under a root. That single restriction makes the graphs straight lines and the algebra clean, which is exactly why CAT leans on it: the difficulty never comes from the manipulation, it comes from reading a dense word problem and translating it into the right equations. Most aspirants lose marks here not because they cannot solve 3x + 5 = 20, but because they set up the wrong relationship, miscount the unknowns, or miss that a "two-variable" story actually gives only one usable equation. This chapter rebuilds the skill from the ground up: solving a single equation cleanly, handling two variables by substitution and elimination, reading off when a pair of equations has a unique, no, or infinite solution from the coefficient ratios, and — the real CAT battleground — turning ages, coins, speeds, work and mixture stories into equations fast. Every section shows the smart setup, the fastest method, and the traps that quietly cost marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- If the question asks only for x + y or x − y, just add or subtract the two equations — never solve for each variable separately.
- Symmetric pair (a x + b y = m, b x + a y = n): adding gives (a + b)(x + y); subtracting gives (a − b)(x − y). Two clean lines, no elimination.
- For solution type, read the ratios at a glance: a₁/a₂ ≠ b₁/b₂ → unique; equal but c differs → none; all equal → infinite.
- Unique-solution shortcut: just check D = a₁b₂ − a₂b₁ ≠ 0; you do not need c at all for "is it unique".
- In word problems, anchor on the smallest unknown and write "is = equals, of = ×, twice = 2×" mechanically to avoid mis-translation.
- Two-digit number = 10t + u; reversing it swaps to 10u + t, and the difference is always 9(t − u) — memorise this for digit problems.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Forgetting a sign when moving a term across the equals sign — the leading cause of wrong one-variable answers.
- Treating a single two-variable equation as if it has one answer; it is a line with infinitely many solutions until a second constraint appears.
- Mixing up the "no solution" and "infinite solution" tests — both need a₁/a₂ = b₁/b₂; the c-ratio decides which.
- In age problems, adding the time gap to only one person instead of to everyone, breaking the constant age difference.
- Misreading "x is 5 less than y" as x − 5 = y instead of x = y − 5, reversing the relationship.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Linear Equations when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
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 |