Linear Equations in Two Variables Notes
An equation of the form ax + by + c = 0 (with a and b not both zero) is a linear equation in two variables. This chapter covers writing such equations, finding their solutions as ordered pairs, drawing their graphs as straight lines, and equations of lines parallel to the axes.
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- 12 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1 key formulas
- 1 concept checks
Linear Equations and Their Solutions
A linear equation in two variables x and y can always be put in the standard form ax + by + c = 0, where a, b, c are real numbers and a and b are not both zero.
A solution is a pair of values (x, y) that makes the equation true. To find solutions, choose any value for one variable and solve for the other. For example, for 2x + y = 6: taking x = 0 gives y = 6; taking x = 1 gives y = 4; taking x = 3 gives y = 0.
A linear equation in two variables has infinitely many solutions, because you can keep choosing new values for x.
Graph of a Linear Equation
Every solution (x, y) of a linear equation in two variables corresponds to a point on its graph, and every point on the graph is a solution. The graph of a linear equation in two variables is a straight line.
To draw the graph: find at least two (preferably three, as a check) solutions, plot the points, and join them with a straight line extended in both directions.
The point where the line meets the x-axis has y = 0, and the point where it meets the y-axis has x = 0. These intercepts are often the easiest points to plot.
Lines Parallel to the Axes
An equation like x = a can be seen as a linear equation in two variables: 1·x + 0·y − a = 0. Its graph is a vertical line, every point of which has x-coordinate a. In one variable, x = a is a single point on the number line; in two variables it is a whole line.
Similarly, y = b has as its graph a horizontal line, every point of which has y-coordinate b.
In particular, the y-axis is the line x = 0 and the x-axis is the line y = 0.
Linear Equations in Two Variables: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Linear Equations and Their Solutions
- Graph of a Linear Equation
- Lines Parallel to the Axes
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (1)
Lines Parallel to the Axes
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
How many solutions does a linear equation in two variables have?
Infinitely many when a and b are not both zero. If b is non-zero, each x determines one y. If b = 0, x is fixed and y can be any real number. Each solution pair is a point on the line.
Why is the graph of ax + by + c = 0 always a straight line?
If b is non-zero, rearrange to y = -(a/b)x - c/b, a straight line with constant slope. If b = 0, then a is non-zero and x = -c/a is a vertical straight line.
Is x = 5 a linear equation in one variable or two?
It depends on context. On the number line it locates one point. In the Cartesian plane it is treated as 1·x + 0·y = 5 and its graph is a vertical line.
How many points are needed to draw the graph of a linear equation?
Two points are enough to fix a line, but it is good practice to plot a third point as a check that all three are collinear.