Pair of Linear Equations in Two Variables — Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 3 · Complete Revision Guide (2026-27)
📌 2026-27 Syllabus Update — Read This First
The Cross-Multiplication Method has been officially removed from the CBSE Class 10 board exam for 2026-27. Only the Graphical Method, Substitution Method, and Elimination Method are examinable this year. If you find old notes or PDFs teaching cross-multiplication as a "third algebraic method," skip that section for board prep — it may still appear in unrelated reference material, but it will not be tested.
1. Standard Form of a Pair of Linear Equations
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Each equation represents a straight line. A "solution" to the pair is the (x, y) point that satisfies both equations simultaneously — geometrically, the point where the two lines meet.
2. Consistency Conditions (Highest Weightage Topic)
| Condition | Type of Lines | Number of Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting | Exactly one (unique solution) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | No solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident | Infinitely many |
⚠ Trap: Mixing up "parallel" vs "coincident"
Both parallel and coincident lines start with a₁/a₂ = b₁/b₂ — students often stop checking there and guess. You must also check the c₁/c₂ ratio: if it breaks the pattern (≠), the lines are parallel (no solution); if it continues the same ratio (=), the lines are coincident (infinite solutions). Never skip the third ratio.
✓ Quick check tip
"Consistent" means the system HAS at least one solution — this covers both the unique-solution and infinite-solutions cases. Only the no-solution case is called "inconsistent." Don't assume "consistent" means only one specific case.
3. Methods of Solving (2026-27 Examinable Methods Only)
A. Graphical Method
Plot both equations as straight lines on the same graph. The coordinates of their intersection point give the solution. If they don't intersect, there's no solution; if they overlap completely, there are infinite solutions.
Plot both equations as straight lines on the same graph. The coordinates of their intersection point give the solution. If they don't intersect, there's no solution; if they overlap completely, there are infinite solutions.
B. Substitution Method
- From one equation, express one variable in terms of the other (e.g., y in terms of x).
- Substitute this expression into the second equation.
- Solve the resulting single-variable equation.
- Substitute back to find the other variable.
Best used when a coefficient of x or y is already 1 or −1 in either equation.
C. Elimination Method
- Multiply one or both equations by suitable constants so that the coefficients of one variable become numerically equal.
- Add or subtract the equations to eliminate that variable.
- Solve the resulting single-variable equation.
- Substitute back into either original equation to find the other variable.
⚠ Trap: Sign error while subtracting equations
When eliminating by subtraction, students often forget to flip the sign of every term in the second equation. Always distribute the subtraction across all terms, not just the variable you're eliminating.
4. Equations Reducible to Linear Form
Some equations look non-linear but can be converted using substitution. For example, in 2/x + 3/y = 4, substitute p = 1/x and q = 1/y to get the linear pair 2p + 3q = 4 (and similarly for the second equation). Solve for p and q first, then find x = 1/p and y = 1/q.
⚠ Trap: Forgetting to convert back
After solving for p and q, students frequently forget the final step of taking the reciprocal to get back to x and y — losing marks even after all the algebra was correct.
5. Word Problem Patterns (Very High Exam Weightage)
| Problem Type | Key Setup |
|---|---|
| Two-digit number problems | Original number = 10x + y; reversed number = 10y + x |
| Boat/stream (upstream-downstream) | Downstream speed = (x+y); Upstream speed = (x−y), where x=boat speed, y=stream speed |
| Ages | Represent present ages as x, y; carefully track "years ago" vs "years hence" as +/− shifts |
| Fixed charge + extra charge | Total = fixed charge + (extra units × rate per unit) |
| Fractions | Represent as x/y; use given conditions after adding/subtracting fixed values to numerator/denominator |
⚠ Trap: Swapping upstream and downstream
A very common error: writing downstream speed as (x−y) and upstream as (x+y) — it's the opposite. Moving with the current (downstream) adds the stream's speed; moving against it (upstream) subtracts it.
⚠ Trap: Reversing digits incorrectly
If the original two-digit number has tens digit x and units digit y, the number itself is 10x + y, not "xy" or "x + y." Students often forget the place-value multiplier.
6. Full Recap — Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Stopping at a₁/a₂ = b₁/b₂ without checking c₁/c₂ | Always check all three ratios to distinguish parallel from coincident |
| "Consistent" = only unique solution | Consistent includes BOTH unique and infinite solution cases |
| Sign errors while subtracting in elimination | Flip the sign of every term in the equation being subtracted |
| Forgetting to convert p, q back to x, y in reducible equations | Always take the reciprocal as the final step |
| Downstream = (x−y), Upstream = (x+y) | It's the reverse: downstream = (x+y), upstream = (x−y) |
| Writing two-digit number as "x+y" or "xy" | Correct form is 10x + y (tens digit × 10, plus units digit) |
| Using Cross-Multiplication Method for boards | Deleted from the 2026-27 CBSE syllabus — use substitution or elimination instead |
Practice What You've Just Revised
Test yourself with the Pair of Linear Equations quiz series once published.
Comments