CBSE Class 10 Mathematics • Chapter 3 • Revision 2026
Pair of Linear Equations in Two Variables Class 10
Revise simultaneous equations with our Pair of Linear Equations in Two Variables Class 10 notes and mind map PDF. The PDF contains notes and a mind map for organising your revision. Use this page to understand graphical, substitution and elimination methods, compare solution conditions and practise worked examples.
Two unknown quantities often need two pieces of information. For example, knowing the total number of pens and pencils gives one equation, while knowing their total cost gives another. Solving both equations together helps you find values that satisfy both conditions.
In Class 10 Maths Pair of Linear Equations in Two Variables, you learn to form equations from situations, represent them as straight lines and solve them algebraically. The key is understanding what a common solution means before choosing a method.
Pair of Linear Equations in Two Variables: Definition
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. Two such equations considered together form a pair of linear equations:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
A solution of the pair is an ordered pair (x, y) satisfying both equations. For example, (3, 2) satisfies x + y = 5 and x − y = 1.
The variables have degree one. Expressions containing x2, xy or y2 are not linear in x and y. A single linear equation represents a straight line and has infinitely many ordered-pair solutions; a pair of lines may have one, none or infinitely many common solutions.
Pair of Linear Equations Class 10 Conditions
Write both equations in the same general form before comparing their coefficients. The familiar ratio conditions below apply when the displayed ratios are defined.
| Coefficient Condition | Graph | Common Solutions | Classification |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting lines | One | Consistent and independent |
| a1/a2 = b1/b2 ≠ c1/c2 | Distinct parallel lines | None | Inconsistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident lines | Infinitely many | Consistent and dependent |
What If a Ratio Has a Zero Denominator?
Do not divide by zero. Instead, compare products or simplify the equations. A unique solution exists when a1b2 − a2b1 ≠ 0. If this difference is zero, check whether one complete equation is a multiple of the other or whether their constants contradict that relationship.
Graphical Method with a Value Table
To solve graphically, find points on each line, plot them using a suitable scale and draw the lines. Their intersection gives the common solution. Consider:
x + y = 5
x − y = 1
| Equation | x | y | Point |
|---|---|---|---|
| x + y = 5 | 0 | 5 | (0, 5) |
| x + y = 5 | 5 | 0 | (5, 0) |
| x − y = 1 | 0 | −1 | (0, −1) |
| x − y = 1 | 1 | 0 | (1, 0) |
The lines intersect at (3, 2). Check this point in both equations: 3 + 2 = 5 and 3 − 2 = 1. Two distinct points determine a straight line; calculating a third point can help check your plotting.
Substitution Method: Solved Example
Example 1: Solve x + y = 5 and 2x − y = 4
From the first equation:
y = 5 − x
Substitute into the second equation:
2x − (5 − x) = 4
2x − 5 + x = 4
3x − 5 = 4
3x = 9
x = 3
Substitute x = 3 into y = 5 − x:
y = 5 − 3
y = 2
Solution: x = 3, y = 2.
Check: 3 + 2 = 5.
Check: 2 × 3 − 2 = 4.
Elimination Method: Solved Example
Example 2: Solve 2x + 3y = 12 and 3x − 2y = 5
Multiply the first equation by 2:
4x + 6y = 24
Multiply the second equation by 3:
9x − 6y = 15
Add the resulting equations:
13x = 39
x = 3
Substitute x = 3 into the first equation:
2 × 3 + 3y = 12
6 + 3y = 12
3y = 6
y = 2
Solution: x = 3, y = 2.
Check in the second equation:
3 × 3 − 2 × 2 = 5.
Should You Add or Subtract?
If the coefficients of the variable you want to eliminate are opposites, add the equations. If they are equal with the same sign, subtract. When subtracting an equation, change the sign of every term in that equation, including the constant.
Word Problems of Pair of Linear Equations
Define your variables clearly before forming equations. Translate each independent condition into a separate equation, solve the pair and interpret the answer in the original situation.
Example 3: Find the Cost of a Pen and a Notebook
Two pens and three notebooks cost ₹90. Three pens and two notebooks cost ₹85. Find the cost of each item.
Let x be the cost of one pen and y the cost of one notebook.
2x + 3y = 90
3x + 2y = 85
Multiply the first equation by 3:
6x + 9y = 270
Multiply the second equation by 2:
6x + 4y = 170
Subtract the second resulting equation from the first:
5y = 100
y = 20
Substitute y = 20:
2x + 3 × 20 = 90
2x + 60 = 90
2x = 30
x = 15
One pen costs ₹15 and one notebook costs ₹20.
Example 4: Recognise No Solution and Infinitely Many Solutions
Consider x + y = 4 and 2x + 2y = 10.
Multiplying the first equation by 2 gives:
2x + 2y = 8
The same expression cannot equal both 8 and 10. The pair has no solution.
Now consider x + y = 4 and 2x + 2y = 8. The second equation is exactly twice the first. They represent the same line and have infinitely many common solutions, including (0, 4), (1, 3) and (2, 2).
Which Solution Method Should You Choose?
- Substitution: Convenient when one variable is already isolated or has coefficient 1 or −1.
- Elimination: Convenient when coefficients already match or can be made equal using small multipliers.
- Graphical method: Useful for understanding the relationship between the lines and required when a question specifically asks for a graph.
Follow any method specified in the question. Otherwise, select the approach that keeps the working simple. Substitution and elimination both reveal a unique solution, a contradiction or an identity.
Pair of Linear Equations Class 10 Mind Map
Use your Pair of Linear Equations in Two Variables Class 10 mind map to connect the general form with three branches: graphical interpretation, algebraic methods and applications. Under graphs, remember intersecting, parallel and coincident lines. Under algebra, connect substitution and elimination to the same common-solution idea.
For your 2026 revision, practise one question from each branch: classify a pair, solve it algebraically, prepare a plotting table and form equations from a word problem. Check every calculated solution in both original equations.
Quick Practice Questions with Answers
- Solve x + y = 9 and x − y = 3. Answer: x = 6, y = 3.
- Solve 2x + y = 7 and x + y = 5. Answer: x = 2, y = 3.
- Classify x + 2y = 6 and 2x + 4y = 12. Answer: Consistent and dependent; infinitely many solutions.
- Classify x + 2y = 6 and 2x + 4y = 15. Answer: Inconsistent; no solution.
- Two numbers have sum 18 and difference 4. Find them. Answer: 11 and 7.
Practise with Your Chapter Worksheet
Apply these methods using the Pair of Linear Equations in Two Variables Class 10 worksheet . Attempt the questions independently before checking your working.
Continue your algebra revision with Quadratic Equations Class 10 notes and mind map , or explore the CBSE Class 10 Maths notes collection .
Frequently Asked Questions
What is a pair of linear equations in two variables?
It is a set of two linear equations involving the same two variables, considered together. A solution must satisfy both equations, not just one.
Why do we usually need two equations to find two unknowns?
One linear equation allows many ordered pairs. A second independent equation provides another condition, which can select one common pair. If the second equation repeats the first condition or contradicts it, there may be infinitely many solutions or none.
Is a dependent pair always consistent?
Yes. A dependent pair represents the same line and therefore has infinitely many common solutions. Consistent means at least one solution exists; it does not mean there must be exactly one.
How do I know whether to use substitution or elimination?
Use substitution when isolating a variable is easy. Use elimination when matching coefficients is easy. Both methods are valid unless the question requires a particular method.
What does 0 = 0 mean while solving equations?
It is an identity. If correct elimination reduces one equation to 0 = 0, the two equations express the same condition. The pair is dependent and has infinitely many solutions. Use the remaining equation to describe those solutions.
What does 0 = 5 mean while solving equations?
It is a contradiction. No values of the variables can make that statement true, so the original pair is inconsistent and has no solution.
Can a coefficient be zero in a linear equation?
Yes, provided both variable coefficients are not zero together. For example, x = 3 represents a vertical line. When a coefficient is zero, avoid undefined ratios and compare products or simplify the equations directly.
How many points are needed to draw each line?
Two distinct points determine a straight line. A third point is useful for checking calculations and plotting accuracy. Use a clear scale and label both axes.
How do I form an equation for a two-digit number?
If x is the tens digit and y is the units digit, the number is 10x + y. Its reversed number is 10y + x. Check that the resulting values are valid digits and satisfy any condition about the reversed number.
Can the solution contain fractions or negative numbers?
Yes. Algebraic solutions need not be positive integers. However, word problems may impose restrictions: a count of notebooks must be a non-negative whole number, for example. Interpret the answer using the original situation.
Why do exercise numbers differ between solution websites?
Textbook editions and boards may use different chapter and exercise numbering. Match the full question with your prescribed textbook rather than relying only on labels such as Exercise 3.2 or 3.3.
What does the notes and mind map PDF contain?
The PDF contains notes and a mind map. Use the notes for concept revision and the mind map for recall, then practise textbook and worksheet questions to improve your solving skills.
What are the most common mistakes in this chapter?
Common errors include changing only some signs during subtraction, multiplying only part of an equation, mixing coefficient signs in consistency tests and checking the answer in only one equation. Write each step clearly and verify both original equations.
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