1. Chapter at a Glance
Many real problems give us two unknowns and two facts: the cost of two items, the speed of a boat and the stream, the ages of two people. Each fact becomes a linear equation in two variables, and together they form a pair. Graphically each equation is a straight line; the solution is where the lines meet. In this chapter you will solve such pairs by the graphical, substitution and elimination methods, decide quickly whether a pair has one, none or infinitely many solutions, and model word problems — a regular source of 3- and 5-mark board questions.
- General form a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0.
- Graphical method and the three possible pictures.
- Consistency conditions using ratios of coefficients.
- Substitution method.
- Elimination method.
- Equations reducible to linear form (e.g. using 1/x, 1/y).
- Word problems: ages, money, digits, fractions, speed, geometry.
2. Graphical Method
Draw both lines (find two or three points for each). Then:
| Picture | Ratios | Solutions | Pair is |
|---|---|---|---|
| Intersecting lines | a₁/a₂ ≠ b₁/b₂ | Exactly one | Consistent |
| Coincident lines | a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many | Dependent (consistent) |
| Parallel lines | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | None | Inconsistent |
(a) 5x − 4y + 8 = 0, 7x + 6y − 9 = 0 (b) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0 (c) 6x − 3y + 10 = 0, 2x − y + 9 = 0
Solution: (a) 5/7 ≠ −4/6 → intersecting, unique. (b) 9/18 = 3/6 = 12/24 = ½ → coincident, infinite. (c) 6/2 = −3/−1 = 3 ≠ 10/9 → parallel, none.
3. Substitution Method
- From one equation, express one variable in terms of the other (choose the simplest).
- Substitute into the other equation; solve the resulting one-variable equation.
- Substitute back to find the second variable. Check in both equations.
Solve 7x − 15y = 2 and x + 2y = 3.
Solution: x = 3 − 2y. 7(3 − 2y) − 15y = 2 → 21 − 29y = 2 → y = 19/29. x = 3 − 38/29 = 49/29. Solution (49/29, 19/29).
4. Elimination Method
- Multiply the equations by suitable numbers so that one variable has equal (or opposite) coefficients.
- Subtract (or add) to eliminate that variable.
- Solve for the remaining variable, then substitute back.
Solve 3x + 4y = 10 and 2x − 2y = 2.
Solution: Multiply the second by 2: 4x − 4y = 4. Add to the first: 7x = 14 → x = 2. Then 2(2) − 2y = 2 → y = 1. (2, 1).
If elimination gives a false statement like 0 = 5 → no solution; a true statement like 0 = 0 → infinitely many.
5. Equations Reducible to Linear Form
Solve 2/x + 3/y = 13 and 5/x − 4/y = −2.
Solution: Let u = 1/x, v = 1/y: 2u + 3v = 13, 5u − 4v = −2. Multiply by 4 and 3: 8u + 12v = 52, 15u − 12v = −6. Add: 23u = 46 → u = 2; v = 3. So x = ½, y = ⅓.
6. Word Problems
| Type | Typical set-up |
|---|---|
| Ages | "x years ago / after": use (x − n), (y + n) |
| Two-digit number | 10x + y; reversed 10y + x |
| Fractions | x/y; adding to numerator/denominator |
| Boats and streams | Downstream u + v, upstream u − v |
| Fixed + variable charges | Total = fixed + rate × quantity |
| Geometry | Angle sums (triangle 180°, cyclic quad opposite 180°), perimeters |
A library charges a fixed amount for the first 3 days and a daily rate after that. Saritha paid ₹27 for 7 days; Susy paid ₹21 for 5 days. Find the fixed charge and daily rate.
Solution: x + 4y = 27, x + 2y = 21 → 2y = 6 → y = 3, x = 15. Fixed ₹15, rate ₹3/day.
In cyclic quadrilateral ABCD, ∠A = 4y + 20, ∠B = 3y − 5, ∠C = −4x, ∠D = −7x + 5 (degrees). Find the angles.
Solution: ∠A + ∠C = 180 → 4y − 4x = 160 → y − x = 40. ∠B + ∠D = 180 → 3y − 7x = 180. Substituting y = x + 40: 3x + 120 − 7x = 180 → x = −15, y = 25. ∠A = 120°, ∠B = 70°, ∠C = 60°, ∠D = 110°.
7. Solved Problems (Board Pattern)
For what k does 3x + y = 1, (2k − 1)x + (k − 1)y = 2k + 1 have no solution?
Solution: 3/(2k − 1) = 1/(k − 1) → 3k − 3 = 2k − 1 → k = 2. Check c-ratio: 1/5 ≠ 1/1 ✓. k = 2.
The sum of the digits of a two-digit number is 9. Nine times the number equals twice the number with digits reversed. Find the number.
Solution: x + y = 9; 9(10x + y) = 2(10y + x) → 88x = 11y → y = 8x. So 9x = 9 → x = 1, y = 8. Number 18.
Places A and B are 100 km apart. Two cars start together from A and B. Moving in the same direction they meet in 5 hours; towards each other they meet in 1 hour. Find their speeds.
Solution: u − v = 20, u + v = 100 → u = 60 km/h, v = 40 km/h.
8. Master Formula Table
| Condition | Result |
|---|---|
| a₁/a₂ ≠ b₁/b₂ | Unique solution (intersecting) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many (coincident) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | No solution (parallel) |
| Two-digit number | 10x + y |
| Downstream / upstream | u + v / u − v |
| Same direction / opposite | relative speed u − v / u + v |
9. Master Summary Table
| Method | Best when | Watch out for |
|---|---|---|
| Graphical | Visual answer, integer solutions | Accuracy with fractions |
| Substitution | A variable has coefficient 1 | Brackets and signs |
| Elimination | Larger coefficients | Multiply every term |
| Ratio test | Number of solutions, find k | Check c-ratio too |
| Reducible form | 1/x, 1/y or 1/(x + y) terms | Convert back at the end |
10. Mind Map
- Intersect/parallel/same
- 1 / 0 / ∞
- Substitution
- Elimination
- a₁/a₂, b₁/b₂, c₁/c₂
- Find k
- Ages, digits, fractions
- Speed, charges
11. Tips and Tricks
- When the x and y coefficients are swapped (ax + by, bx + ay), add and subtract the equations.
- Always verify the answer in both original equations — it takes 20 seconds and saves marks.
- For graph questions, choose points with integer coordinates and label the intersection clearly.
- For "infinitely many solutions", checking only a₁/a₂ = b₁/b₂ and forgetting c₁/c₂.
- Sign errors when subtracting equations.
- In reducible equations, giving u and v as the final answer instead of x and y.
12. Important One-liners
- Each linear equation in two variables represents a straight line.
- A consistent pair has at least one solution.
- Parallel lines ⇒ inconsistent pair ⇒ no solution.
- Coincident lines ⇒ infinitely many solutions.
- Elimination giving 0 = 0 means the equations are dependent.
13. Exam-pattern Questions with Detailed Answers
The pair x + 2y = 3, 2x + 4y = 7 has:
- a unique solution
- no solution
- infinitely many solutions
- exactly two solutions
Show answer
1/2 = 2/4 ≠ 3/7 → parallel lines.
Answer: (B)
Solve x + y = 14, x − y = 4.
Show answer
Add: 2x = 18 → x = 9, y = 5.
Answer: (9, 5)
Find k if kx + 3y = 3 and 12x + ky = 6 have infinitely many solutions.
Show answer
k/12 = 3/k → k² = 36 → k = ±6. Also 3/k = 3/6 → k = 6. (k = −6 fails: 3/−6 ≠ 3/6.)
Answer: k = 6
Solve by elimination: 3x − 5y − 4 = 0, 9x = 2y + 7.
Show answer
3x − 5y = 4 (×3): 9x − 15y = 12. 9x − 2y = 7. Subtract: −13y = 5 → y = −5/13. 3x = 4 − 25/13 = 27/13 → x = 9/13.
Answer: x = 9/13, y = −5/13
5 pencils and 7 pens cost ₹50; 7 pencils and 5 pens cost ₹46. Find the cost of each.
Show answer
Add: 12(x + y) = 96 → x + y = 8. Subtract: 2(y − x) = 4 → y − x = 2. Pencil ₹3, pen ₹5.
Answer: Pencil ₹3, pen ₹5
A fraction becomes 1 if 1 is added to the numerator and subtracted from the denominator; it becomes ½ if 1 is added to the denominator. Find the fraction.
Show answer
x + 1 = y − 1 → x − y = −2. 2x = y + 1 → 2x − y = 1. Subtract: x = 3, y = 5.
Answer: 3/5
Ten years hence, A will be twice as old as B. Five years ago, A was three times as old as B. Find their present ages.
Show answer
x + 10 = 2(y + 10) → x − 2y = 10. x − 5 = 3(y − 5) → x − 3y = −10. Subtract: y = 20, x = 50.
Answer: A 50, B 20
The perimeter of a rectangle is 40 cm and its length is 4 cm more than its breadth. Find its dimensions.
Show answer
l + b = 20, l − b = 4 → l = 12, b = 8.
Answer: 12 cm × 8 cm
Solve 10/(x + y) + 2/(x − y) = 4 and 15/(x + y) − 5/(x − y) = −2.
Show answer
Let p = 1/(x + y), q = 1/(x − y): 10p + 2q = 4, 15p − 5q = −2. Multiply first by 5, second by 2: 50p + 10q = 20, 30p − 10q = −4. Add: 80p = 16 → p = 1/5; q = (4 − 2)/2 = 1. So x + y = 5, x − y = 1 → x = 3, y = 2.
Answer: x = 3, y = 2
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km downstream. Find the speed of the stream and of the boat in still water.
Show answer
Let p = 1/(u − v), q = 1/(u + v): 30p + 44q = 10, 40p + 55q = 13. Solving: q = 1/11, p = 1/5 → u + v = 11, u − v = 5 → u = 8, v = 3.
Answer: Boat 8 km/h, stream 3 km/h
Draw x − y + 1 = 0 and 3x + 2y − 12 = 0. Find the vertices of the triangle these lines form with the x-axis, and its area.
Show answer
Intersection: y = x + 1 → 3x + 2x + 2 = 12 → x = 2, y = 3 → (2, 3). x-intercepts: (−1, 0) and (4, 0). Base 5, height 3 → area 7.5 square units.
Answer: (2, 3), (−1, 0), (4, 0); area 7.5 sq units
For which values of a and b does 2x + 3y = 7, (a + b)x + (2a − b)y = 21 have infinitely many solutions?
Show answer
2/(a + b) = 3/(2a − b) = 7/21 = 1/3. So a + b = 6 and 2a − b = 9 → 3a = 15 → a = 5, b = 1.
Answer: a = 5, b = 1
14. Quick Revision Summary
- Two lines: intersect (1 solution), parallel (none), coincide (infinitely many).
- Ratio test with a₁/a₂, b₁/b₂, c₁/c₂ decides the case.
- Substitution and elimination give exact solutions; always verify.
- Use u = 1/x, v = 1/y for reducible equations.
- Model ages, digits, fractions, speeds and charges carefully.