CBSE Class 10 Mathematics • Chapter 3
Pair of Linear Equations in Two Variables Class 10 Worksheet
Practise solving simultaneous equations, interpreting graphs and converting word problems into equations. Revise the main formulas, compare solution conditions and check your understanding with worked examples, MCQs and additional practice questions.
This Pair of Linear Equations in Two Variables Class 10 worksheet supports practice in one of the central topics of Class 10 Maths. The chapter connects algebra with everyday situations involving prices, ages, numbers and quantities. To solve these problems, you need to identify two unknowns, form two equations and find the values that satisfy both.
Use the worksheet available on this page alongside the revision material below. These explanations and original examples help you understand substitution, elimination and the graphical method. Attempt the questions first, then check each calculation and substitute your answer into both original equations.
Pair of Linear Equations in Two Variables: Definition and Examples
A linear equation in two variables has the general form ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero. Examples include x + y = 7 and 2x − y = 5. Its graph is a straight line.
A pair of linear equations in two variables consists of two such equations involving the same variables. A solution of the pair is an ordered pair (x, y) that satisfies both equations simultaneously.
A Simple Pair of Linear Equations Example
x + y = 7
x − y = 1
The values x = 4 and y = 3 satisfy both equations:
4 + 3 = 7
4 − 3 = 1
Therefore, (4, 3) is the solution of this pair. A value that satisfies only one equation is not a solution of the pair.
Pair of Linear Equations in Two Variables Class 10 Formulas
Write both equations in the same standard form before comparing coefficients:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The following conditions connect the coefficients with the number of solutions and the position of the two lines.
| Condition | Graph | Solutions | Classification |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting lines | One unique solution | Consistent and independent |
| a1/a2 = b1/b2 ≠ c1/c2 | Distinct parallel lines | No solution | Inconsistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident lines | Infinitely many solutions | Consistent and dependent |
Use Ratios Only When Their Denominators Are Non-Zero
If a denominator is zero, avoid dividing by it. Instead, compare equivalent equations or use cross-products. For example, a1b2 − a2b1 ≠ 0 guarantees a unique solution.
For x = 2 and y = 3, some coefficient ratios are undefined, but the equations clearly have the unique solution (2, 3).
Methods to Solve a Pair of Linear Equations
Graphical Method
Prepare a table of values for each equation and plot the two lines. Their intersection gives the common solution. Parallel lines have no common solution; coincident lines have infinitely many.
Substitution Method
Express one variable in terms of the other using one equation. Substitute that expression into the second equation, solve and then find the remaining variable.
Elimination Method
Make the coefficients of one variable equal or opposite. Add or subtract the equations to eliminate that variable, then solve for the other.
Verification
Substitute the calculated values into both original equations. This checks the solution and helps catch sign errors or incorrect multiplication.
Class 10 Pair of Linear Equations: Solved Examples
Example 1: Solve by Substitution
x + y = 9
2x − y = 6
From the first equation:
y = 9 − x
Substitute into the second equation:
2x − (9 − x) = 6
2x − 9 + x = 6
3x − 9 = 6
3x = 15
x = 5
Substitute x = 5 into y = 9 − x:
y = 9 − 5
y = 4
Answer: x = 5, y = 4.
Check the first equation: 5 + 4 = 9.
Check the second equation: 2(5) − 4 = 6.
Example 2: Solve by Elimination
2x + 3y = 17 … (1)
3x − 2y = 6 … (2)
Multiply equation (1) by 2:
4x + 6y = 34 … (3)
Multiply equation (2) by 3:
9x − 6y = 18 … (4)
Add equations (3) and (4):
13x = 52
x = 4
Substitute x = 4 into equation (1):
2(4) + 3y = 17
8 + 3y = 17
3y = 9
y = 3
Answer: x = 4, y = 3.
Check: 2(4) + 3(3) = 17.
Check: 3(4) − 2(3) = 6.
Example 3: Identify an Inconsistent Pair
x + 2y = 5
2x + 4y = 12
Multiplying the first equation by 2 gives:
2x + 4y = 10
The second equation requires the same expression to equal 12. Both statements cannot be true simultaneously.
Answer: The pair has no solution. The lines are distinct and parallel, so the pair is inconsistent.
Graphical Method: Pair of Linear Equations Class 10 Table
Consider x + y = 6 and x − y = 2. Prepare the following plotting table. Each ordered pair in a row satisfies the corresponding equation.
| Equation | Point 1 | Point 2 | Point 3 |
|---|---|---|---|
| x + y = 6 | (0, 6) | (4, 2) | (6, 0) |
| x − y = 2 | (0, −2) | (2, 0) | (4, 2) |
Plot the points on the same coordinate plane and draw the two straight lines. They intersect at (4, 2), giving x = 4 and y = 2. Two distinct points determine a line; a third point helps check your plotting.
Word Problems of Pair of Linear Equations in Two Variables
Case-Based Question: Pens and Notebooks
A student pays ₹80 for two pens and three notebooks. Another student pays ₹70 for four pens and one notebook. Find the price of each item.
Let the price of one pen be ₹x.
Let the price of one notebook be ₹y.
Form the equations:
2x + 3y = 80 … (1)
4x + y = 70 … (2)
Multiply equation (2) by 3:
12x + 3y = 210 … (3)
Subtract equation (1) from equation (3):
10x = 130
x = 13
Substitute x = 13 into equation (2):
4(13) + y = 70
52 + y = 70
y = 18
Answer: One pen costs ₹13 and one notebook costs ₹18.
Check the first purchase:
2(13) + 3(18) = 80
Check the second purchase:
4(13) + 18 = 70
For age problems, add or subtract the same number of years from both present ages. For a two-digit number with tens digit x and units digit y, write the number as 10x + y and its reversal as 10y + x. Defining the variables clearly makes the equations easier to form.
Pair of Linear Equations in Two Variables Class 10 MCQ
MCQ 1: Number of Solutions
The equations x + y = 4 and 2x + 2y = 8 have:
A. No solution
B. One unique solution
C. Infinitely many solutions
D. Exactly two solutions
Show Answer
Answer: C. The second equation is twice the first. Both represent the same line, so there are infinitely many solutions.
MCQ 2: Find the Common Solution
What is the solution of x + y = 8 and x − y = 2?
A. (3, 5)
B. (5, 3)
C. (4, 4)
D. (6, 2)
Show Answer
Answer: B. (5, 3).
Adding the equations gives 2x = 10, so x = 5.
Substitution gives y = 3.
MCQ 3: Identify Parallel Lines
The equations 3x + 2y = 7 and 6x + 4y = 9 represent:
A. Intersecting lines
B. Coincident lines
C. Distinct parallel lines
D. Perpendicular lines
Show Answer
Answer: C. Doubling the first equation gives 6x + 4y = 14, which contradicts the second equation. The pair has no solution.
MCQ 4: Form a Word Problem Equation
If a pen costs ₹x and a notebook costs ₹y, which equation represents three pens and two notebooks costing ₹66?
A. 2x + 3y = 66
B. 3x + 2y = 66
C. 3x − 2y = 66
D. x + y = 66
Show Answer
Answer: B. Three pens cost ₹3x and two notebooks cost ₹2y. Their total is ₹66.
Class 10 Pair of Linear Equations Practice Questions
Use these original questions as a short assignment or revision test. They cover calculation, consistency and reasoning.
- Solve x + y = 11 and x − y = 3.
- Solve 2x + y = 13 and x − y = 2 by substitution.
- Solve 3x + 2y = 18 and x + 2y = 10 by elimination.
- Determine the number of solutions of 2x + 3y = 6 and 4x + 6y = 12.
- Find k for which 2x + 3y = 7 and 4x + ky = 15 have no solution.
- The sum of two numbers is 27 and their difference is 5. Form two equations and find the numbers.
Check the Practice Answers
- x = 7, y = 4.
- x = 5, y = 3.
- x = 4, y = 3.
- Infinitely many solutions; the second equation is twice the first.
- k = 6. The left sides are proportional, but the constants are not.
- x + y = 27 and x − y = 5; the numbers are 16 and 11.
Revision Tips and Common Errors
- Move all terms to the same side before comparing coefficients.
- Multiply every term, including the constant, when scaling an equation.
- Use brackets when subtracting an entire equation.
- Check whether the question asks for a solution, a graph or only the number of solutions.
- In word problems, state the meaning and units of x and y.
- Verify the answer in both original equations.
Pair of Linear Equations Class 10 Notes, Mind Map and Solutions
FAQs on Pair of Linear Equations in Two Variables Class 10
What is a pair of linear equations in two variables?
It is a set of two linear equations involving the same two variables. For example, x + y = 7 and x − y = 1 form a pair. Their common solution is x = 4 and y = 3 because these values satisfy both equations.
What is the difference between one linear equation and a pair?
One linear equation in two variables represents a straight line with infinitely many points. A pair requires the same ordered pair to satisfy both equations. Depending on the lines, the pair may have one solution, no solution or infinitely many solutions.
What are the conditions for a unique solution, no solution and infinitely many solutions?
When the coefficient ratios are defined, unequal ratios of the x and y coefficients give a unique solution. Equal x and y coefficient ratios but a different constant ratio give no solution. If all three ratios are equal, there are infinitely many solutions. The conditions table above shows each case.
How many solutions does a consistent pair have?
A consistent pair has at least one solution. Two intersecting lines give one unique solution, while coincident lines give infinitely many solutions. Consistent does not always mean a unique solution.
What does an inconsistent pair of equations mean?
An inconsistent pair has no common solution. For example, x + y = 3 and x + y = 5 cannot both be satisfied by the same values. Their graphs are distinct parallel lines.
When should I use substitution or elimination?
Substitution is convenient when a variable can be isolated easily, especially when its coefficient is 1 or −1. Elimination is convenient when coefficients are already equal or opposite, or can be made so using small multipliers. Both methods give the same common solution.
What do 0 = 0 and 0 = 5 mean when solving a pair?
For a valid pair of line equations, an identity such as 0 = 0 after elimination indicates dependent equations with infinitely many solutions. A contradiction such as 0 = 5 indicates an inconsistent pair with no solution.
How do I solve a pair of linear equations graphically?
Find at least two distinct points for each equation, plot them on the same coordinate plane and draw the lines. Read their intersection point as the solution. If the lines are distinct and parallel, there is no solution; if they coincide, there are infinitely many.
How do I form equations from word problems?
Define the two unknown quantities first. Translate each independent statement into an equation using those variables. For example, if two numbers have sum 20 and difference 4, write x + y = 20 and x − y = 4. Solve and interpret the answer in the original context.
Can a pair of linear equations have exactly two solutions?
No. Two straight lines either intersect at one point, remain distinct and parallel, or coincide. Therefore, a pair has one solution, no solution or infinitely many solutions.
Which topics should I practise for Class 10 revision and PYQs?
Practise solution conditions, graphical interpretation, substitution, elimination, unknown coefficient questions and word problems. Use these skills when attempting Class 10 Pair of Linear Equations PYQs from verified papers. The original examples on this page are additional practice and are not labelled as past board questions.
Are Practice Sets 1.1 to 1.5 the same as CBSE Chapter 3 exercises?
No. Maharashtra State Board and CBSE NCERT textbooks use different chapter and exercise numbering. If you need Linear Equations in Two Variables Practice Sets 1.1, 1.2, 1.3, 1.4, 1.5 or Problem Set 1, refer to the Maharashtra Board Class 10 Maths 1 Chapter 1 Solutions . For CBSE exercises such as 3.1, 3.2 or 3.3, follow your NCERT edition.
What is the chapter weightage in the Class 10 board exam?
Do not assume a fixed mark total for this individual chapter in every paper. Use the official syllabus, assessment information and sample paper for your board and examination year. Prepare both direct equation-solving questions and applications.
Where can I find Class 10 linear equations notes and a mind map?
Visit the Pair of Linear Equations in Two Variables Notes and Mind Map on WitKnowLearn. Review the concepts there, then return to this Grade 10 worksheet for practice.
What is a pair of linear equations in two variables called in Hindi?
इसे हिंदी में दो चरों वाले रैखिक समीकरणों का युग्म कहते हैं। उदाहरण के लिए, x + y = 7 और x − y = 1 एक युग्म है। x = 4 और y = 3 दोनों समीकरणों को संतुष्ट करते हैं, इसलिए (4, 3) इस युग्म का हल है।
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