Chapter 4 · NCERT Class 9 Mathematics

Linear Equations in Two Variables

An equation like 2x + y = 7 links two unknowns instead of one. It has not a single answer but infinitely many — a whole line of solutions (x, y). Learn the form ax + by + c = 0, how to find solutions, and how the equation draws a straight line. Tap each idea to see what it means.

📐 3 topics⏱ ~25 min📝 12-question quiz

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Reading a linear equation in two variables

Take 2x + y = 7. It has its own vocabulary — the form, its solutions, its graph. Tap each idea to see what it points to and how the pieces fit together.

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The three big ideas

A linear equation in two variables has the form ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero.
It links two unknowns, x and y, each to the first power — no x², no 1/x, no √x.
Examples: 2x + y = 7, x − 3y = 2 and y = 5 (here a = 0) are all linear equations in two variables.
A single such equation has infinitely many solutions, not just one.

Common mistake: a linear equation in two variables has infinitely many solutions — a whole line of pairs (x, y), not a single answer like a one-variable equation (2x + 3 = 0 gives just x = −1.5).

A solution is an ordered pair (x, y) that makes the equation true when substituted in.
To find one: choose any value of x, put it in the equation, and solve for y.
Because you can pick infinitely many values of x, there are infinitely many solutions.

Worked example. Find one solution of 2x + y = 7.

Step 1. Choose a value of x — let x = 1.

Step 2. Substitute: 2(1) + y = 7 ⇒ 2 + y = 7.

Step 3. Solve for y: y = 7 − 2 = 5.

So (1, 5) is a solution. (Pick x = 2 instead and you get (2, 3) — another of the infinitely many.)

Plot a few solutions on the coordinate plane and join them — the graph is always a straight line.
Every point on the line is a solution, and every solution lies on the line.
Lines parallel to the axes: x = a is a vertical line (parallel to the y-axis); y = b is a horizontal line (parallel to the x-axis).
Two points are enough to draw the line — a third makes a handy check.

Where you'll meet it

Linear equations, at work

Relating two quantities

If sweets cost ₹5 each, the total cost y for x sweets is y = 5x — a linear equation in two variables. Each pair (x, y) is one purchase, and the graph shows how cost grows steadily with the number of items.

Linear unit conversions

Converting Celsius to Fahrenheit follows F = (9/5)C + 32 — a straight-line relation between two variables. Reading temperatures, currency or distances often means using such a linear equation and its graph.