Chapter 3: Pair of Linear Equations in Two Variables

1. Core Concepts

General Form: A pair of linear equations in two variables ($x$ and $y$) is represented as:

$a_1x + b_1y + c_1 = 0$

$a_2x + b_2y + c_2 = 0$

Nature of Solutions (The Table Rule):

Intersecting Lines: If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system has a Unique Solution (Consistent).

Coincident Lines: If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system has Infinitely Many Solutions (Dependent/Consistent).

Parallel Lines: If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the system has No Solution (Inconsistent).

Algebraic Methods to Solve:

Substitution Method: Express one variable in terms of the other from one equation and substitute it into the second.

Elimination Method: Multiply equations by suitable numbers to make the coefficients of one variable equal, then add or subtract to eliminate it.


2. Important Examples

Example 1: Solve for $x$ and $y$ using Elimination Method:

$x + y = 5$ and $2x - 3y = 4$

Solution:

Multiply the first equation by 3: $3x + 3y = 15$.

Add this to the second equation: $(3x + 3y) + (2x - 3y) = 15 + 4$.

$5x = 19 \Rightarrow x = 19/5$.

Substitute $x$ in the first equation: $19/5 + y = 5 \Rightarrow y = 5 - 19/5 = 6/5$.

Result: $x = 3.8, y = 1.2$.


3. Previous Year Questions (PYQs) with Solutions

Q1. (1 Mark) For what value of $k$ will the equations $x + 2y = 3$ and $5x + ky + 7 = 0$ represent parallel lines?

Solution: For parallel lines, $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

$1/5 = 2/k$

Result: $k = 10$.

Q2. (3 Marks) The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits differ by 2, find the number.

Solution:

Let the digits be $x$ and $y$. Number = $10x + y$. Reversed = $10y + x$.

Case 1: $(10x + y) + (10y + x) = 66 \Rightarrow 11x + 11y = 66 \Rightarrow x + y = 6 = 0$ (Eq 1).

Case 2: $x - y = 2$ (Eq 2).

Solving Eq 1 & 2: $2x = 8 \Rightarrow x = 4$ and $y = 2$.

Result: The number is 42 (or 24).

Q3. (5 Marks) 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.

Solution:

Let pencil cost be $x$ and pen cost be $y$.

$5x + 7y = 50$ (Eq 1)

$7x + 5y = 46$ (Eq 2)

Multiply Eq 1 by 7 and Eq 2 by 5:

$35x + 49y = 350$

$35x + 25y = 230$

Subtracting: $24y = 120 \Rightarrow y = 5$.

Put $y=5$ in Eq 1: $5x + 35 = 50 \Rightarrow 5x = 15 \Rightarrow x = 3$.

Result: Cost of 1 pencil = ₹3, Cost of 1 pen = ₹5.