Class 9 Notes - Linear Equations in Two Variables

1. Introduction to Linear Equations in Two Variables

A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers, and x and y are variables. The highest power of the variables is 1.

2. Standard Form

The general form is ax + by + c = 0.
Examples: 2x + 3y = 5, x - y + 4 = 0

3. Solution of a Linear Equation in Two Variables

A solution is a pair of values (x, y) that satisfies the equation.
There are infinitely many solutions for such equations.
Example: For x + y = 5, (2, 3) and (4, 1) are solutions.

4. Graph of a Linear Equation in Two Variables

The graph is always a straight line.
Every point on the line is a solution to the equation.
To draw the graph, find at least two solutions (points), plot them, and join with a straight line.

Example: For x + y = 4, points (0,4), (4,0), and (2,2) all satisfy the equation. Plot these points and draw a line through them.

5. Intercepts

x-intercept: The point where the line crosses the x-axis (set y = 0).
y-intercept: The point where the line crosses the y-axis (set x = 0).

Example: For 2x + 3y = 6:
x-intercept: Set y = 0 ⇒ 2x = 6 ⇒ x = 3 ⇒ (3, 0)
y-intercept: Set x = 0 ⇒ 3y = 6 ⇒ y = 2 ⇒ (0, 2)

6. Applications

Word problems involving two unknowns (age, money, distance, etc.)
Representing real-life situations as equations and finding solutions graphically or algebraically.

7. Practice Problems

Find three solutions of the equation 2x + y = 7.
Plot the graph of x - 2y = 4.
Find the x- and y-intercepts of the equation 3x + 2y = 12.
If (1, k) is a solution of 3x + 2y = 11, find the value of k.
Write a linear equation in two variables whose graph passes through the point (2, -3).

8. Summary

A linear equation in two variables has infinitely many solutions.
Its graph is a straight line.
Every point on the line is a solution to the equation.
Intercepts help in drawing the graph easily.