Chapter 3: Pair of Linear Equations in Two Variables
1. Basic Definitions
An equation of the form ax + by + c = 0, where a, b, c are real numbers and a and b are not both zero, is a linear equation in two variables x and y. Its graph is always a straight line, and every point on the line is a solution.
A pair of linear equations in two variables is:
a2x + b2y + c2 = 0
where a12 + b12 ≠ 0 and a22 + b22 ≠ 0
A solution is a pair (x, y) that satisfies both equations simultaneously — geometrically, a point lying on both lines.
2. Graphical Representation: Three Possibilities
Two lines in a plane can:
- Intersect at one point → exactly one (unique) solution → pair is consistent.
- Be parallel → no solution → pair is inconsistent.
- Be coincident (same line) → infinitely many solutions → pair is consistent (dependent).
3. Conditions on Coefficients (Most Important Table)
| Condition | Graph | Number of solutions | Nature |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting lines | Exactly one (unique) | Consistent |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel lines | No solution | Inconsistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident lines | Infinitely many | Consistent (dependent) |
Micro-example: For x + 2y − 4 = 0 and 2x + 4y − 12 = 0: a1/a2 = 1/2, b1/b2 = 2/4 = 1/2, c1/c2 = −4/−12 = 1/3. Since 1/2 = 1/2 ≠ 1/3, the lines are parallel ⇒ no solution.
4. Graphical Method of Solution
- From each equation, express y in terms of x (or make a table of at least 2–3 ordered pairs for each line).
- Plot the points for each equation and draw the two lines on the same axes.
- Read the point of intersection — its coordinates give the solution. If lines are parallel: no solution; if coincident: infinitely many.
5. Algebraic Methods
(a) Substitution Method — Steps
- From one equation, express one variable in terms of the other (say y = f(x)).
- Substitute this expression into the other equation to get an equation in one variable.
- Solve it to find the value of that variable.
- Substitute back to find the second variable. Write the solution as (x, y).
Worked example: Solve x + y = 14, x − y = 4.
From (2): x = y + 4. Put in (1): y + 4 + y = 14 ⇒ 2y = 10 ⇒ y = 5. Then x = 9. Solution: x = 9, y = 5.
(b) Elimination Method — Steps
- Multiply one or both equations by suitable non-zero constants to make the coefficients of one variable numerically equal.
- Add or subtract the equations to eliminate that variable.
- Solve the resulting one-variable equation.
- Substitute the value in either original equation to get the other variable.
Worked example: Solve 2x + 3y = 8 and 4x − y = 2.
Multiply (2) by 3: 12x − 3y = 6. Add to (1): 14x = 14 ⇒ x = 1. From (2): 4(1) − y = 2 ⇒ y = 2. Solution: (1, 2).
6. Word Problems — Standard Types
Translate the situation into two equations, solve, and state the answer with units. Common patterns:
- Numbers/digits: two-digit number = 10x + y; number obtained by reversing digits = 10y + x.
- Ages: "5 years ago", "10 years hence" — adjust both ages by the same amount.
- Fractions: let the fraction be x/y; form equations from the given changes to numerator/denominator.
- Fixed + variable charges: hostel charges, taxi fare (fixed charge x + per-km charge y).
- Speed, distance, time: boat/stream problems — downstream speed = (u + v), upstream speed = (u − v), where u = speed of boat in still water, v = speed of stream. Use time = distance/speed.
- Cost problems: cost of pencils/pens, tickets, etc.
Micro-example (ages): Aftab tells his daughter, "Seven years ago I was seven times as old as you. Three years from now I shall be three times as old as you." Equations: x − 7 = 7(y − 7) and x + 3 = 3(y + 3), i.e., x − 7y + 42 = 0 and x − 3y − 6 = 0. Solving: y = 12, x = 42.
7. Summary of Key Points
- Every solution of a linear equation in two variables is a point on its line.
- Consistency is decided purely by the ratios a1/a2, b1/b2, c1/c2.
- Both algebraic methods (substitution and elimination) must give the same unique solution when a1/a2 ≠ b1/b2.