Chapter 3: Pair of Linear Equations in Two Variables

Important questions — the ones the board asked most (20)

1. Q1Long answer5 marksAsked: 2026

   Find the coordinates of the vertices of the triangle formed by the graph, whose sides are represented by the equations 2x - 3y + 6 = 0; 2x + 3y - 18 = 0 and x = 0. Also find the area of the triangle.

   

   Medha Ma’am

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   Model answer · graphical method, triangle vertices

   1. 1.Line 1: 2x - 3y + 6 = 0, so y = (2x + 6)/3. Table of two or more solutions: x: 0, 3, -3 y: 2, 4, 0 Plot the points A(0, 2), B(3, 4) and (-3, 0) on graph paper. Join them to get the line of 2x - 3y + 6 = 0.1 mark
   2. 2.Line 2: 2x + 3y - 18 = 0, so y = (18 - 2x)/3. Table of solutions: x: 0, 3, 9 y: 6, 4, 0 Plot the points C(0, 6), B(3, 4) and (9, 0) on the same graph paper. Join them to get the line of 2x + 3y - 18 = 0.1 mark
   3. 3.Line 3: x = 0 is the y-axis. Every point on it has x-coordinate 0, for example (0, 2) and (0, 6).0.5 marks
   4. 4.The three lines meet in pairs at the following points. Line 1 and Line 2 intersect at B(3, 4). Line 1 meets x = 0 at A(0, 2). Line 2 meets x = 0 at C(0, 6). So the coordinates of the vertices of the triangle are A(0, 2), B(3, 4) and C(0, 6).1.5 marks
   5. 5.The base AC lies along the y-axis. Base AC = 6 - 2 = 4 units. The height is the distance of B from the y-axis, which is its x-coordinate: 3 units. Area of triangle ABC = 1/2 × base × height = 1/2 × 4 × 3 = 6 sq. units.1 mark

   Diagram: On graph paper with the x-axis and y-axis marked, draw the lines 2x - 3y + 6 = 0 (through (-3, 0), A(0, 2), B(3, 4)), 2x + 3y - 18 = 0 (through C(0, 6), B(3, 4), (9, 0)) and x = 0 (the y-axis). Label each line with its equation, mark the vertices A(0, 2), B(3, 4), C(0, 6) and shade triangle ABC.

   • Write a table of at least two solutions for each line before plotting, and label each line on the graph with its equation.
   • State all three vertices clearly: "A(0, 2), B(3, 4) and C(0, 6)".
   • Show "1/2 × 4 × 3" and finish with the unit "6 sq. units".

   Marks split as in the board’s published marking scheme

2. Q2Long answer5 marksAsked: 2026

   Two sections, A and B, of class X contributed a total of 1500 for the Uttarakhand flood victims. The contribution from X-A was 100 less than that of X-B. Graphically, find the amounts contributed by both sections.

   

   Medha Ma’am

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   Model answer · Graphical solution, word problem

   1. 1.Let x and y be the amounts (in Rs) contributed by the classes X-A and X-B respectively. According to the question, the total contribution is Rs 1500, so x + y = 1500 ... (1) The contribution from X-A is Rs 100 less than that of X-B, so x = y − 100, i.e., y − x = 100 ... (2)1 mark
   2. 2.Graph of x + y = 1500, i.e., y = 1500 − x. Two or more solutions of this equation are: x: 0, 700, 1500 y: 1500, 800, 0 Plot the points A(0, 1500), C(700, 800) and B(1500, 0) on graph paper and join them to form the line AB, which represents x + y = 1500.1.5 marks
   3. 3.Graph of y − x = 100, i.e., y = x + 100. Two or more solutions of this equation are: x: 0, 700, −100 y: 100, 800, 0 Plot the points P(0, 100), C(700, 800) and Q(−100, 0) on the same graph paper and join them to form the line PQ, which represents y − x = 100. The lines AB and PQ intersect at the point C(700, 800), so x = 700, y = 800 is the unique solution of the pair of linear equations.1.5 marks
   4. 4.Therefore, the amount contributed by class X-A = x = Rs 700.0.5 marks
   5. 5.And the amount contributed by class X-B = y = Rs 800. (Check: 700 + 800 = 1500 and 800 − 700 = 100.)0.5 marks

   Diagram: On one graph paper, with a suitable scale (for example 1 cm = Rs 100) on the x-axis (amount from X-A) and the y-axis (amount from X-B), draw line AB through A(0, 1500) and B(1500, 0), labelled x + y = 1500, and line PQ through P(0, 100) and Q(−100, 0), labelled y − x = 100. Mark and label the point where they cross as C(700, 800).

   • Begin with "Let x and y be the amounts contributed by the classes X-A and X-B respectively" before you write the two equations.
   • Give a table of at least two points for each equation, and write each equation as a label on its line in the graph.
   • Mark the point where the lines cross, C(700, 800), and write both final amounts with the unit: Rs 700 for X-A and Rs 800 for X-B.

   Marks split as in the board’s published marking scheme

3. Q3Long answer5 marksAsked: 2026

   Draw the graph of the pair of linear equations x - y + 2 = 0 and 4x - y - 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 135 s

   Model answer · Graphical method, triangle area

   1. 1.Graph of 4x – y – 4 = 0: this gives y = 4x – 4. We find solutions of the equation: x = 0 gives y = –4 x = 1 gives y = 0 x = 2 gives y = 4 We plot the points P(0, –4), B(1, 0) and A(2, 4) on graph paper and join them. This straight line represents 4x – y – 4 = 0, and it cuts the x-axis at B(1, 0).2 marks
   2. 2.Graph of x – y + 2 = 0: this gives y = x + 2. We find solutions of the equation: x = –2 gives y = 0 x = 0 gives y = 2 x = 2 gives y = 4 We plot the points C(–2, 0), Q(0, 2) and A(2, 4) on the same graph paper and join them. This straight line represents x – y + 2 = 0, and it cuts the x-axis at C(–2, 0). The two lines intersect at the point A(2, 4), so x = 2, y = 4 is the unique solution, and the pair of equations is consistent.2 marks
   3. 3.The two lines and the x-axis form triangle ABC with vertices A(2, 4), B(1, 0) and C(–2, 0). Base BC = 1 – (–2) = 3 units Height = distance of A from the x-axis = 4 units Area of triangle ABC = 1/2 × base × height = 1/2 × 3 × 4 = 6 sq. units1 mark

   Diagram: On one graph paper, draw the line through P(0, –4), B(1, 0) and A(2, 4) and label it 4x – y – 4 = 0. Draw the line through C(–2, 0), Q(0, 2) and A(2, 4) and label it x – y + 2 = 0. Mark the x-axis, the y-axis and the origin O, and shade triangle ABC.

   • Write the table of x and y values for each equation, and label each line with its equation on the graph.
   • Write the vertices of the triangle: A(2, 4), B(1, 0) and C(–2, 0).
   • End with "Area of triangle ABC = 1/2 × 3 × 4 = 6 sq. units" and include the unit.

   Marks split as in the board’s published marking scheme

4. Q4Long answer5 marksAsked: 2026

   Aarush bought 2 pencils and 3 chocolates for Rs 11 and Tanish bought 1 pencil and 2 chocolates for Rs 7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.

   How the model answer begins

   1. Let the cost of 1 pencil be ₹ x and the cost of 1 chocolate be ₹ y. Aarush bought 2 pencils and 3 chocolates for ₹ 11, so 2x + 3y = 11 ...(1)

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5. Q5Long answer5 marksAsked: 2026

   Aarush bought 2 pencils and 3 chocolates for 11 and Tanish bought 1 pencil and 2 chocolates for 7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.

   How the model answer begins

   1. Let the price of 1 pencil be ₹x and the price of 1 chocolate be ₹y. Aarush bought 2 pencils and 3 chocolates for ₹11, so the first equation is: 2x + 3y = 11 ... (1)

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6. Q6Long answer5 marksAsked: 2026

   Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair : x - 5y = 6; 2x - 10y = 12

   How the model answer begins

   1. For the first equation x - 5y = 6, two solutions are found by choosing values of x and finding the corresponding y. When x = 6, y = 0, and when x = 1, y = -1. So the solutions are the coordinates (6, 0) and (1, -1).

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7. Q7Long answer5 marksAsked: 2026

   (a) Solve the following system of equations graphically : x - 2y = 3, 3x - 8y = 7 OR (b) Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. To know the present ages of Adil and Bharat : (i) form the linear equations representing the above information. (ii) show that the system of equations is consistent with unique solution. (iii) find the present ages of Adil and Bharat.

8. Q8Long answer5 marksAsked: 2026

   (a) Solve the following system of equations graphically : 2x + 3y = 5, -3x + y = -2 OR (b) The sum of the digits of a 2-digit number is 11. The number obtained by interchanging its digits exceeds the given number by 9. To know the number : (i) form the linear equations representing the above situation. (ii) verify that the equations have a unique solution. (iii) solve the equations to get the given 2-digit number.

9. Q9Long answer5 marksAsked: 2026

   35. (a) The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. (i) Representing the above situation, frame two linear equations in two variables. (ii) Show that the equations have unique solution. (iii) Solve the equations and hence find the numbers. OR (b) Solve the following equations graphically : x + y = 7 and 2x - 5y = 7

   How the model answer begins

   1. (b) The given pair of linear equations is x + y = 7 ... (1) and 2x – 5y = 7 ... (2). To draw the graph of each equation, we find two or more solutions of each equation. For Equation (1), y = 7 – x: x: 0, 7, 6 y: 7, 0, 1…

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10. Q10Long answer5 marksAsked: 2025

    Determine graphically whether the following pair of linear equations 2x + 3y = 12 and x - y = 1 has unique solution or infinitely many solutions.

    How the model answer begins

    1. To draw the graphs, we find solutions of each equation. For 2x + 3y = 12, y = (12 – 2x)/3: x: 0, 3, 6 y: 4, 2, 0 So the points are A(0, 4), B(3, 2) and C(6, 0). For x – y = 1, y = x – 1: x: 0, 1, 3 y: –1, 0, 2 So the…

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11. Q11Long answer5 marksAsked: 2025

    The sum of a 2-digit number and the number obtained by reversing the order of its digits, is 121. The two digits differ by 3. (i) Represent the above information in the form of pair of linear equations. (ii) Show that the equations have unique solution. (iii) Solve the equations and find the number.

    How the model answer begins

    1. Let the unit's digit of the number be y and the ten's digit be x, where x > y.

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12. Q12Long answer5 marksAsked: 2025

    34. Solve the following pair of equations using graphical method : 3x - 4y + 3 = 0 and -2x + 5y = 9

13. Q13Long answer5 marksAsked: 2025

    (a) Solve the following pair of linear equations by graphical method : 2x + y = 9 and x - 2y = 2 OR (b) Nidhi received simple interest of Rs 1,200 when invested Rs x at 6% p.a. and Rs y at 5% p.a. for 1 year. Had she invested Rs x at 3% p.a. and Rs y at 8% p.a. for that year, she would have received simple interest of Rs 1,260. Find the values of x and y.

    How the model answer begins

    1. The given pair of linear equations is 2x + y = 9 (1) and x − 2y = 2 (2). To draw the graph of each equation, we find solutions of each equation. From (1), y = 9 − 2x: x: 0, 2, 4 y: 9, 5, 1 Points: A(0, 9), B(2, 5), C(4,…

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14. Q14Long answer5 marksAsked: 2025

    (a) Solve the following pair of linear equations by graphical method : 2x + y = 9 and x - 2y = 2 OR (b) Nidhi received simple interest of ₹ 1,200 when invested ₹ x at 6% p.a. and ₹ y at 5% p.a. for 1 year. Had she invested ₹ x at 3% p.a. and ₹ y at 8% p.a. for that year, she would have received simple interest of ₹ 1,260. Find the values of x and y.

    How the model answer begins

    1. Answering part (a). Given pair of linear equations: 2x + y = 9 ... (1) and x − 2y = 2 ... (2). To draw the graph of each equation, find solutions of each equation. Equation (1): y = 9 − 2x x: 0, 2, 4 y: 9, 5, 1 Points:…

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15. Q15Long answer5 marksAsked: 2024

    (a) Using graphical method, solve the following equations : 3x + y + 4 = 0 and 3x - y + 2 = 0 OR (b) Tara scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each wrong answer, then Tara would have scored 50 marks. Assuming that Tara attempted all questions, find the total number of questions in the test.

    How the model answer begins

    1. Solving option (b). Given: Tara scored 40 marks when she got 3 marks for each right answer and lost 1 mark for each wrong answer. She would have scored 50 marks if she got 4 marks for each correct answer and lost 2…

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16. Q16Long answer5 marksAsked: 2024

    (a) Using graphical method, solve the following pair of equations : x + 2y = 8 and 3x - 2y = 12 OR (b) The sum of the digits of a 2-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

    How the model answer begins

    1. Given: the pair of linear equations x + 2y = 8 and 3x − 2y = 12. To use the graphical method, we find a few solutions of each equation. For x + 2y = 8, y = (8 − x)/2: x: 0, 2, 4, 8 y: 4, 3, 2, 0 For 3x − 2y = 12, y =…

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17. Q17Long answer5 marksAsked: 2024

    The ratio of monthly incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save 2,000 per month, then find their monthly incomes.

    How the model answer begins

    1. Given: the ratio of the monthly incomes of two persons is 9 : 7, the ratio of their expenditures is 4 : 3, and each person saves ₹2000 per month. Let the monthly incomes of the two persons be ₹9x and ₹7x.

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18. Q18Long answer5 marksAsked: 2024

    If 2 is added to the numerator of a fraction and 1 is added to the denominator, it becomes 3/4. If 2 is subtracted from the numerator and 1 is subtracted from the denominator, it becomes 1/2. Find the fraction.

    How the model answer begins

    1. The fraction asked for is 7/11. The working below shows how we get it. Given: adding 2 to the numerator and 1 to the denominator makes the fraction 3/4. Subtracting 2 from the numerator and 1 from the denominator makes…

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19. Q19Long answer5 marksAsked: 2024

    The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

    How the model answer begins

    1. The required two-digit number is 18. It is found as follows. Given: the sum of the digits of a two-digit number is 9, and nine times the number is twice the number obtained by reversing its digits. Let the digit at…

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20. Q20Long answer5 marksAsked: 2025

    Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received ₹ 1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received ₹ 20 more as annual interest. How much money did he invest in each scheme ?

    How the model answer begins

    1. Vijay invested ₹ 12,000 in scheme A and ₹ 10,000 in scheme B. The working is shown below. Given: scheme A gives 8% per annum and scheme B gives 9% per annum. The total annual interest is ₹ 1,860. Let Vijay invest ₹ x in…

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67 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has :

   • (A) a unique solution
   • (B) two solutions
   • (C) no solution
   • (D) an infinite number of solutions ✓

   Answer: (D)

2. 2. CBSE 2026

   If the pair of linear equations : a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is consistent and dependent, then

   • (A) a1/a2 != b1/b2
   • (B) a1/a2 != b1/b2 = c1/c2
   • (C) a1/a2 = b1/b2 != c1/c2
   • (D) a1/a2 = b1/b2 = c1/c2 ✓

   Answer: (D)

3. 3. CBSE 2026

   The pair of linear equations 3x/2 + 5y/3 = 7 and 9x + 10y = 14, is :

   • (A) consistent
   • (B) inconsistent ✓
   • (C) consistent with one solution
   • (D) consistent with many solutions

   Answer: (B)

4. 4. CBSE 2026

   Equation of another line parallel to the line represented by 2x - 6y = 7 is :

   • (A) y = 3x - 7
   • (B) 2x = 9 - 6y
   • (C) x - 3y = 7 ✓
   • (D) x = 7/2 - 3y

   Answer: (C)

5. 5. CBSE 2026

   The equation of the coincident line represented by 5x - 2y = 3 is :

   • (A) 5x - 4y = 3
   • (B) 5x - 4y + 6 = 0
   • (C) 15x - 12y - 3 = 0
   • (D) 5x - 4y - 6 = 0 ✓

   Answer: (D)

6. 6. CBSE 2026

   Assertion (A) : The system of linear equations 3x - 5y + 7 = 0 and -6 x + 10 y + 14 = 0 is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

   • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
   • (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false. ✓
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (C)

7. 7. CBSE 2025

   If (0, 0) is the solution of the equation x + y = c - 1, then the value of c is :

   • (A) 0
   • (B) 1 ✓
   • (C) -1
   • (D) any real number

   Answer: (B)

8. 8. CBSE 2025

   The number of solutions of the system of equations x = 3, y = -1 is :

   • (A) 0
   • (B) 1 ✓
   • (C) 2
   • (D) Infinite

   Answer: (B)

9. 9. CBSE 2025

   The number of solutions of the system of equations x = a, x = b (a != b) is :

   • (A) 0 ✓
   • (B) 1
   • (C) 2
   • (D) infinitely many

   Answer: (A)

10. 10. CBSE 2025

    For what value(s) of k does the system of equations kx + 2y = 3 and 2x + y = 5 have a unique solution ?

    • (A) k = a real number
    • (B) k ≠ 8
    • (C) k ≠ 4 ✓
    • (D) k = 4

    Answer: (C)

Questions parents and students ask

How many marks does "Pair of Linear Equations in Two Variables" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 5 marks and about 2 questions a year; by marks it ranks 10 in Maths.

Which are the most important (VVI) questions of "Pair of Linear Equations in Two Variables"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 87 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.