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One equation in one variable, like 2x = 6, has one answer. One equation in two variables, like 2x + 3y = 12, has infinitely many, and that surprises students. This chapter explains why, turns the solutions into a graph, and then (new for Class 9 this year) solves two equations together. Here are the ideas, worked examples you can check, and the mistakes to avoid.
What does this chapter cover in 2026-27?
In NCERT's new Class 9 textbook, this is Chapter 13, Two Variables, One Line, in Ganita Manjari Part II. It defines the standard form ax + by + c = 0 (with a and b not both zero), the slope-intercept form, and then a pair of linear equations. The CBSE Class IX curriculum for 2026-27 lists graphical solutions, slope-intercept form, a pair of linear equations, the graphical method, consistency and inconsistency, and substitution and elimination. It sits in the Algebra unit, which carries 20 of the 80 theory marks. In the 2025-26 CBSE syllabus a pair of linear equations was a Class 10 chapter, so this is a real change.
Why does one equation have infinitely many solutions?
Take 2x + 3y = 12. A solution is any pair (x, y) that makes it true.
x = 0 gives 3y = 12, so (0, 4).
y = 0 gives 2x = 12, so (6, 0).
x = 3 gives 6 + 3y = 12, so y = 2 and (3, 2).
You can pick any x and solve for y, so there is no end to the solutions. Plot them and they fall on one straight line. Every point on that line is a solution, and every solution is a point on that line.
What is the slope-intercept form?
Rearrange to make y the subject. From 2x + 3y = 12: 3y = −2x + 12, so y = (−2/3)x + 4.
The number in front of x, −2/3, is the slope: for every 3 units you move right, the line drops 2.
The constant, 4, is where the line crosses the y-axis.
This works whenever b is not zero. If b = 0 you get something like x = 3, a vertical line parallel to the y-axis. That is still a linear equation in two variables: it is 1·x + 0·y − 3 = 0, and every point (3, y) satisfies it.
How do you solve a pair of linear equations?
Two lines usually cross at one point, and that point satisfies both equations.
Worked example 1 (elimination): x + y = 7 and 2x − y = 5.
Add the equations: the y terms cancel, giving 3x = 12, so x = 4.
Substitute into the first: 4 + y = 7, so y = 3.
Check in the second: 2(4) − 3 = 5. Correct. The lines meet at (4, 3).
Worked example 2 (substitution, a word problem): a school sells 20 tickets for a play. Adult tickets cost ₹50 and child tickets ₹30, and the total is ₹760. How many of each?
Let a be adult tickets and c child tickets. Then a + c = 20 and 50a + 30c = 760.
From the first, c = 20 − a. Substitute: 50a + 30(20 − a) = 760, so 20a + 600 = 760 and a = 8.
Then c = 12. Check: 50 × 8 + 30 × 12 = 400 + 360 = 760.
Can a pair of equations have no solution?
Yes. Compare slopes.
3x + 4y = 7 and 6x + 8y = 14: the second is the first multiplied by 2. They are the same line, so there are infinitely many solutions. The pair is consistent.
3x + 4y = 7 and 6x + 8y = 10: same slope (−3/4) but different intercepts (7/4 and 5/4). The lines are parallel and never meet. No solution, so the pair is inconsistent.
Lines with different slopes always cross once: one solution.
When elimination makes both variables vanish, look at what is left. "0 = 0" means the same line; "0 = 4" means parallel lines.
Which mistakes lose marks?
Sign errors when subtracting equations. Subtracting 2x − y from x + y gives −x + 2y, not −x. Write the subtraction in brackets.
Not checking in both equations. A solution that fits one equation only is not a solution of the pair.
Uneven graph scales, which make the crossing point hard to read. Choose a scale where your intercepts fit.
Reading a graphical answer as exact. If lines cross at about (1.7, 2.3), confirm with algebra.
A practice routine for this chapter
A review by Dunlosky and colleagues (2013) rated practice testing and distributed practice as the most useful study techniques it assessed. So spread your practice out and test yourself.
Session 1: graph three single equations, plotting three points each, and write each in slope-intercept form.
Session 2 (two days later): solve four pairs, two by substitution and two by elimination, and check each in both equations.
Session 3 (a week later): write one word problem yourself, solve it, and classify three pairs as one, none or infinitely many solutions before solving.
If you are stuck setting up a word problem, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For plotting basics, see our guide to coordinate geometry in Class 9, and for the older Class 10 treatment, pairs of linear equations in Class 10.
Graph 2x + 3y = 12 tonight with three points and check that (3, 2) lands on the line.
