Linear Equations in Two Variables
Concept-first revision with intuition, formal rules, worked examples, exam reasoning and focused practice.
Chapter Snapshot
- A linear equation in two variables can be written as ax+by+c=0, with a and b not both zero.
- A solution is an ordered pair that satisfies the equation.
- A linear equation in two variables generally has infinitely many solutions.
- The graph of all its solutions is a straight line.
- Equations x=a and y=b represent lines parallel to the coordinate axes.
- A table of values is a bridge between an equation and its graph.
What You Will Learn
- Recognise a linear equation in two variables
- Understand what a solution means
- Generate multiple ordered-pair solutions
- Plot solutions and obtain a straight-line graph
- Interpret equations such as x=a and y=b
- Model simple real situations using a linear equation
Why This Chapter Matters
In one-variable equations, the goal was to find a single number. In two variables, one equation usually cannot determine both variables uniquely. Instead, it describes a relationship between them.
For example, x+y=10 is satisfied by (1,9), (2,8), (7,3) and infinitely many other pairs. Each pair is a solution, and when those solutions are plotted, they line up on a straight line.
This is the first major meeting point of algebra and coordinate geometry. An equation is not just something to solve; it is also a geometric object.
Bridge from Earlier Classes
Before starting, be comfortable with:
- Linear equations in one variable
- Ordered pairs
- Cartesian plane
- Substitution
These are the tools this chapter reuses and deepens.
1. Form of a Linear Equation
A linear equation in two variables can be written as ax+by+c=0, where a, b and c are real numbers and a and b are not both zero.
The word linear means each variable appears only to the first power and variables are not multiplied together. So x²+y=5 and xy=6 are not linear equations in two variables.
2. What Is a Solution?
A solution is an ordered pair (x,y) that makes the equation true. For x+y=5, (2,3) is a solution because 2+3=5, while (2,4) is not.
The same pair may satisfy one equation and fail another. Always substitute both coordinates into the actual equation instead of judging from appearance.
3. Why Infinitely Many Solutions?
One equation imposes one condition on two variables. If one variable is chosen freely, the equation usually determines the other.
For x+y=5, choose x=0 and y=5; choose x=1 and y=4; choose x=2.5 and y=2.5. There is no end to such choices. This is why the solution set is a line rather than one point.
4. From Equation to Table
To graph an equation, choose convenient values for one variable and calculate the other. Record the ordered pairs in a table.
Two distinct points determine a straight line, but using a third point is a valuable check. If the third point fails to lie on the same line, recheck arithmetic or plotting.
5. Graph of a Linear Equation
Every solution lies on the graph, and every point on the graph represents a solution. This equivalence means a point can be tested either algebraically by substitution or geometrically by checking whether it lies on the line.
The mathematical line extends indefinitely, although a real-life context may restrict meaningful values.
6. Special Equations x=a and y=b
The equation x=a fixes the horizontal coordinate but allows any y-value, giving a vertical line parallel to the y-axis.
The equation y=b fixes the vertical coordinate but allows any x-value, giving a horizontal line parallel to the x-axis. These cases show clearly what each coordinate controls.
7. Modelling Real Situations
If notebooks cost ₹20 and pens ₹10 and total spending is ₹100, then 20x+10y=100 models the situation.
The equation has infinitely many real solutions, but the shopping context usually requires x and y to be non-negative whole numbers. Context therefore filters mathematically valid solutions into practically meaningful ones.
Visualise the Core Idea
A linear equation in two variables is represented by all points on a straight line.
Formula / Theorem Sheet
Formula / Key Relations
General form
a and b are not both zero.
Vertical line
Parallel to the y-axis.
Horizontal line
Parallel to the x-axis.
Worked Examples — Follow the Reasoning
Yes, so (2,3) is a solution.
If y=0, x=6 → (6,0).
If y=1, x=4 → (4,1).
If y=3, x=0 → (0,3).
How to Think in the Exam
Treat the equation as a relationship. Generate solutions systematically, plot carefully, and remember that the whole straight line—not just the points in your table—is the solution set.
Common Mistakes
Key Terms
Quick Revision
Quick Revision
- A linear equation in two variables can be written as ax+by+c=0, with a and b not both zero.
- A solution is an ordered pair that satisfies the equation.
- A linear equation in two variables generally has infinitely many solutions.
- The graph of all its solutions is a straight line.
- Equations x=a and y=b represent lines parallel to the coordinate axes.
- A table of values is a bridge between an equation and its graph.
- Revise the reasoning behind each rule once, not only the final formula.
Focused Practice
Which is linear in x and y: x²+y=3, xy=4, 2x-3y=7?
View Solution
2x-3y=7
Check whether (1,2) satisfies x+2y=5.
View Solution
Yes.
Give two solutions of x-y=1.
View Solution
For example (1,0) and (3,2).
Write an equation if x ₹10 items and y ₹5 items cost ₹50 in total.
View Solution
10x+5y=50
Why do two distinct points determine the graph of a linear equation?
View Solution
Because its graph is a straight line, and exactly one straight line passes through two distinct points.
Final Chapter Summary
Chapter Summary
- A linear equation in two variables can be written as ax+by+c=0, with a and b not both zero.
- A solution is an ordered pair that satisfies the equation.
- A linear equation in two variables generally has infinitely many solutions.
- The graph of all its solutions is a straight line.
- Equations x=a and y=b represent lines parallel to the coordinate axes.
- A table of values is a bridge between an equation and its graph.
- Strong revision means understanding why the method works, then practising enough examples to recognise when to use it.