The Fourth Chapter of CBSE Class 9 Maths “Linear Equations in Two Variables”

Concepts

1. Linear Equation in Two Variables:
   A linear equation in two variables is an equation of the form: ax+by+c=0ax + by + c = 0 where:
   • a,b,a, b, and cc are real numbers.
   • xx and yy are variables.
   Example: 2x+3y−5=02x + 3y – 5 = 0.

2. Solution of a Linear Equation in Two Variables:
   A solution is a pair of values (x,y)(x, y) that satisfy the equation. For example, for 2x+3y−5=02x + 3y – 5 = 0:
   Substituting x=1,y=1x = 1, y = 1: 2(1)+3(1)−5=0(True).2(1) + 3(1) – 5 = 0 \quad \text{(True)}.

3. Graphical Representation:
   • A linear equation in two variables represents a straight line on a graph.
   • To draw the graph:
     1. Rewrite the equation in the form y=mx+cy = mx + c (slope-intercept form).
     2. Find at least two solutions.
     3. Plot the solutions on the graph.
     4. Join the points to form a straight line.

Solved Examples

Example 1: Plotting the Graph

Problem: Plot the graph of 3x+2y=63x + 2y = 6.

Solution:
Rewrite as y=−32x+3y = \frac{-3}{2}x + 3.
Find solutions by substituting values of xx:

Plot points (0,3)(0, 3), (2,0)(2, 0), and (4,−3)(4, -3), then join them to form a line.

Example 2: Checking Solutions

Problem: Check whether (1,2)(1, 2) is a solution of 2x−y+3=02x – y + 3 = 0.

Solution: Substitute x=1,y=2x = 1, y = 2: 2(1)−2+3=0(False).2(1) – 2 + 3 = 0 \quad \text{(False)}.

Answer: (1,2)(1, 2) is not a solution.

Example 3: Word Problem

Problem: A taxi company charges a fixed rate of ₹50 plus ₹15 per kilometer. Write a linear equation and find the cost of traveling 10 km.

Solution:

• Let yy be the cost and xx be the distance (in km).
• Equation: y=15x+50.y = 15x + 50.
• For x=10x = 10: y=15(10)+50=150+50=200.y = 15(10) + 50 = 150 + 50 = 200.

Answer: Cost = ₹200.

Applications

1. Economics:
   Linear equations model relationships between costs and revenues, such as profit/loss analysis.
2. Physics:
   Used to represent uniform motion or constant acceleration scenarios.
3. Business and Finance:
   Helps determine supply-demand relationships, fixed costs, and variable costs.
4. Computer Programming:
   Algorithms use linear equations for graphical representations and optimizations.

COMMENTS ON YOUR QUESTION REGARDING LINEAR EQUATIONS..