Linear Equations in Two Variables
A linear equation in two variables, such as 2x + 3y = 6, has not one but infinitely many solutions, and its graph is a straight line. This Class 9 chapter teaches the standard form of such equations, how to find and check solutions, how to draw their graphs, and the special equations of lines parallel to the axes.
Learning objectives
- Write a linear equation in two variables in standard form.
- Find solutions of a linear equation as ordered pairs.
- Understand that such an equation has infinitely many solutions.
- Draw the graph of a linear equation.
- Write equations of lines parallel to the x- and y-axes.
Key concepts
Standard form and solutions
A linear equation in two variables can be written as ax + by + c = 0, where a and b are not both zero. A solution is an ordered pair (x, y) that makes the equation true; there are infinitely many such pairs.
Graph of a linear equation
Plotting the solutions of a linear equation gives points that all lie on a single straight line. So the graph of ax + by + c = 0 is a line, and every point on that line is a solution.
Lines parallel to the axes
The equation x = a (a constant) is a vertical line parallel to the y-axis, and y = b is a horizontal line parallel to the x-axis. In two variables, x = 2 means all points with x-coordinate 2.
Checking a solution
To test whether a pair (x, y) is a solution, substitute the values into the equation; if both sides are equal, the pair is a solution.
Important formulas
Standard form
ax + by + c = 0 (a, b not both 0)
Vertical line
x = a (parallel to the y-axis)
Horizontal line
y = b (parallel to the x-axis)
Key definitions
- Linear equation in two variables
- An equation of the form ax + by + c = 0 with a, b not both zero.
- Solution
- An ordered pair (x, y) that satisfies the equation.
- Graph of an equation
- The set of all points whose coordinates satisfy the equation — a straight line here.
Solved examples
Q1. Find two solutions of 2x + y = 7.
Solution: Put x = 0: y = 7, giving (0, 7). Put x = 1: 2 + y = 7 ⇒ y = 5, giving (1, 5). So (0, 7) and (1, 5) are two solutions.
Q2. Is (2, 3) a solution of x + y = 5?
Solution: Substitute: 2 + 3 = 5, which is true, so (2, 3) is a solution.
Q3. What does the graph of x = 3 look like?
Solution: It is a straight line parallel to the y-axis, passing through all points with x-coordinate 3, such as (3, 0), (3, 1) and (3, −2).
Common mistakes to avoid
- Thinking a linear equation in two variables has only one solution.
- Writing x = 3 as a single point instead of a vertical line in the plane.
- Plotting too few points or a wrong point, giving a crooked 'line'.
- Mixing up which axis x = a and y = b are parallel to.
Linear Equations in Two Variables — MCQ Quiz
11 questions with instant feedback. Use number keys 1–4 to answer.
A linear equation in two variables has:
Practice questions
Short answer
Write any one solution of x − y = 0.
(2, 2) — since 2 − 2 = 0 (any pair with x = y works).
What is the equation of the y-axis?
x = 0.
How many solutions does 5x + 2y = 10 have?
Infinitely many.
Long answer
Draw the graph of x + y = 4 by finding three solutions.
Solutions: (0, 4), (4, 0) and (2, 2). Plotting these three points and joining them gives a straight line — the graph of x + y = 4. Each point on the line is a solution of the equation.
Express 'the cost of a pen (x) and a pencil (y) is ₹20' as a linear equation and give two solutions.
x + y = 20. Two solutions: (10, 10) — pen ₹10, pencil ₹10; and (15, 5) — pen ₹15, pencil ₹5.
HOTS (Higher Order Thinking)
The graph of a linear equation passes through (0, 0). What can you say about its constant term c?
Substituting (0, 0) into ax + by + c = 0 gives c = 0, so the line passes through the origin only when c = 0.
Lines x = 3 and y = 2 are drawn on the same plane. At which point do they meet?
x = 3 is vertical, y = 2 is horizontal; they intersect at the single point (3, 2).
Quick revision
Revision notes
- Standard form: ax + by + c = 0 (a, b not both 0).
- A solution is an ordered pair; there are infinitely many.
- Graph is a straight line; two points are enough to draw it.
- x = a is vertical (∥ y-axis); y = b is horizontal (∥ x-axis).
Key takeaways
- Two variables ⇒ infinitely many solutions ⇒ a line.
- Find easy solutions by setting x = 0 and y = 0.
- Remember x-axis is y = 0 and y-axis is x = 0.
Frequently asked questions
Why does a linear equation in two variables have infinitely many solutions?
Because for every value of x you choose, you can find a matching y, so there are endless ordered pairs that satisfy it.
How do I draw the graph of such an equation?
Find at least two solution points, plot them, and join them with a straight line.
What is the equation of a line parallel to the x-axis?
It has the form y = b, where b is a constant.