Chapter 3: Pair of Linear Equations in Two Variables

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

1. Q1Long answer5 marksAsked: 2020, 2022

   Draw the graph of the pair of linear equations x + y = 7 and 2x - 3y = 9 and solve it.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 135 s

   Model answer · graphical solution, pair of linear equations

   1. 1.For the equation x + y = 7, we find at least two solutions. x + y = 7 can be written as y = 7 - x. When x = 0, y = 7. When x = 7, y = 0. When x = 2, y = 5. So the solutions (x, y) of x+y=7 are (0, 7), (7, 0) and (2, 5).1 mark
   2. 2.For the equation 2x - 3y = 9, we find at least two solutions. 2x - 3y = 9 can be written as x = (9 + 3y)/2. When y = 1, x = 6. When y = -1, x = 3. When y = 3, x = 9. So the solutions (x, y) of 2x - 3y = 9 are (6, 1), (3, -1) and (9, 3).1 mark
   3. 3.We plot the points (0, 7), (7, 0) and (2, 5) on the graph paper and draw a straight line passing through them to represent the equation x + y = 7. We plot the points (6, 1), (3, -1) and (9, 3) on the same graph paper and draw a straight line passing through them to represent the equation 2x - 3y = 9. Both the lines are drawn on the same axes with the x-axis and y-axis marked, each point labelled with its coordinates.2 marks
   4. 4.The two lines intersect at the point (6, 1). So, x = 6, y = 1 is the required solution of the pair of linear equations x + y = 7 and 2x - 3y = 9.1 mark

   Diagram: Draw the x-axis and y-axis on graph paper; plot and label points (0,7), (7,0), (2,5) and draw the line for x+y=7, plot and label points (6,1), (3,-1), (9,3) and draw the line for 2x-3y=9, mark the intersection point as (6, 1).

   • Write the table of at least two solutions for each equation before drawing the graph, showing x+y=7 and 2x-3y=9 clearly.
   • Label both axes and every plotted point with its coordinates on the graph.
   • State the final answer as the coordinates of the intersection point, writing x = 6, y = 1.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Short answer2 marksAsked: 2020, 2022

   Find the solution of the equations 8x + 5y = 9 and 3x + 2y = 4 by cross-multiplication method.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 130 s

   Model answer · cross-multiplication method, linear equations

   1. 1.Write 8x + 5y − 9 = 0 and 3x + 2y − 4 = 0: a1 = 8, b1 = 5, c1 = −9, a2 = 3, b2 = 2, c2 = −4. Cross-multiplication: x/(b1c2 − b2c1) = y/(c1a2 − c2a1) = 1/(a1b2 − a2b1).1 mark
   2. 2.x/(5 × (−4) − 2 × (−9)) = y/((−9) × 3 − (−4) × 8) = 1/(8 × 2 − 3 × 5), so x/(−2) = y/5 = 1/1. Hence x = −2, y = 5. (Check: 8(−2) + 5(5) = 9; 3(−2) + 2(5) = 4.)1 mark

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Long answer5 marksAsked: 2025

   Draw the graphs of the pair of linear equations x + 3y - 6 = 0 and 2x - 3y - 12 = 0 and solve them.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 160 s

   Model answer · graphical solution, pair of linear equations

   1. 1.The given equation x + 3y - 6 = 0 can be written as x + 3y = 6. To draw its graph, we find at least two solutions of this equation. When x = 0, 3y = 6, so y = 2. When y = 0, x = 6. Thus we get two solutions of the equation x + 3y = 6, namely (0, 2) and (6, 0).1 mark
   2. 2.The given equation 2x - 3y - 12 = 0 can be written as 2x - 3y = 12. To draw its graph, we find at least two solutions of this equation. When x = 6, 2(6) - 3y = 12, so -3y = 0, giving y = 0. When x = 0, -3y = 12, so y = -4. Thus we get two solutions of the equation 2x - 3y = 12, namely (6, 0) and (0, -4).1 mark
   3. 3.We plot the points A(0, 2) and B(6, 0) on the graph paper and join them to form the line AB, which represents the equation x + 3y - 6 = 0. We also plot the points B(6, 0) and C(0, -4) on the same graph paper and join them to form the line BC, which represents the equation 2x - 3y - 12 = 0.1.5 marks
   4. 4.On plotting both lines on the graph paper, we observe that there is a point common to both the lines AB and BC. This common point, where the line representing x + 3y - 6 = 0 and the line representing 2x - 3y - 12 = 0 intersect, is B(6, 0).1 mark
   5. 5.Since (6, 0) is the only common point on both the lines, the solution of the given pair of linear equations x + 3y - 6 = 0 and 2x - 3y - 12 = 0 is x = 6 and y = 0.0.5 marks

   Diagram: Draw two perpendicular axes (x-axis and y-axis) on graph paper; plot points A(0, 2) and B(6, 0) and join them to draw line AB labelled "x + 3y - 6 = 0"; plot points B(6, 0) and C(0, -4) and join them to draw line BC labelled "2x - 3y - 12 = 0"; mark the intersection point B as (6, 0).

   • Write the working showing x = 0, y = 2 and y = 0, x = 6 for x + 3y = 6.
   • Write the working showing x = 6, y = 0 and x = 0, y = -4 for 2x - 3y = 12.
   • State clearly that the intersection point is (6, 0) and hence x = 6, y = 0 is the solution.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Long answer5 marksAsked: 2025

   A two digit number is four times of the sum of its digits and twice the product of its digits. Find the number.

   How the model answer begins

   1. Let the ten's digit of the required two-digit number be x and the unit's digit be y. Then the first number may be written in the expanded form as 10x + y.

   The full answer from Medha Ma’am →
5. Q5Long answer5 marksAsked: 2026

   Plot the graphs of the pair of linear equations 2x - 3y + 13 = 0 and 3x - 2y + 12 = 0 and solve.

   How the model answer begins

   1. The given pair of linear equations is 2x - 3y + 13 = 0 and 3x - 2y + 12 = 0. To represent these equations graphically, we need at least two solutions for each equation. For 2x - 3y + 13 = 0: When x = -5, 2(-5) - 3y + 13…

   The full answer from Medha Ma’am →
6. Q6Long answer5 marksAsked: 2026

   Draw the graphs of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Find the coordinates of the vertices of the triangle formed by these lines and the x-axis.

   How the model answer begins

   1. For the equation x - y + 1 = 0, we write y = x + 1 and find at least two solutions. We prepare a table of values: x = 0, y = 1 x = -1, y = 0 These two points (0, 1) and (-1, 0) satisfy x - y + 1 = 0 and are used to draw…

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7. Q7Long answer5 marksAsked: 2026

   Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

   How the model answer begins

   1. Let the speed of Ritu in still water be x km/h and the speed of the stream be y km/h.

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8. Q8Long answer5 marksAsked: 2022

   The sum of the digits of a two-digit number is 15. On reversing the digits, the number obtained is 9 more than the original number. Find the number.

   How the model answer begins

   1. Let the unit digit of the two-digit number be x and the tens digit be y. So the first number may be written as 10y + x in the expanded form. When the digits are reversed, y becomes the unit's digit and x becomes the…

   The full answer from Medha Ma’am →
9. Q9Long answer5 marksAsked: 2022

   Using graphical method, solve the pair of equations 4x - 5y = 20 and 3x + 5y = 15.

   How the model answer begins

   1. For the equation 4x - 5y = 20, we find two solutions. Putting x = 0, we get -5y = 20, i.e., y = -4. Putting y = 0, we get 4x = 20, i.e., x = 5. So we get the points (5, 0) and (0, -4). Table: x 5 0; y 0 -4.

   The full answer from Medha Ma’am →
10. Q10Long answer5 marksAsked: 2022

    Draw the graphs of pair of linear equations x - 4y + 14 = 0 and 3x + 2y - 14 = 0 and solve them.

    How the model answer begins

    1. For the equation x - 4y + 14 = 0, we need at least two solutions. Writing it as x = 4y - 14, we find the solutions by choosing values of y. When y = 2, x = 8 - 14 = -6. When y = 4, x = 16 - 14 = 2. When y = 0, x = -14.…

    The full answer from Medha Ma’am →
11. Q11Long answer5 marksAsked: 2023

    Using graphical method solve the pair of equations 3x - 5y + 1 = 0 and 2x - y - 3 = 0.

    How the model answer begins

    1. Given equation (1): 3x - 5y + 1 = 0, so 5y = 3x + 1, which gives y = (3x+1)/5. When x = -2, y = (-6+1)/5 = -1. When x = 3, y = (9+1)/5 = 2. When x = 8, y = (24+1)/5 = 5. So three solutions of equation (1) are (-2, -1),…

    The full answer from Medha Ma’am →
12. Q12Long answer5 marksAsked: 2023

    Using graphical method solve the pair of equations 3x - 5y - 1 = 0 and 2x - y - 3 = 0.

    How the model answer begins

    1. The first equation is 3x - 5y - 1 = 0, which can be written as 3x - 5y = 1. To represent this equation graphically, we need at least two solutions. We write y = (3x-1)/5. When x = 2, y = (3(2)-1)/5 = 5/5 = 1. When x =…

    The full answer from Medha Ma’am →
13. Q13Long answer5 marksAsked: 2021

    Draw the graph of the pair of linear equations x - 2y = 5 and 2x + 4y = 10 and solve.

    How the model answer begins

    1. For the equation x - 2y = 5, we find at least two solutions. Writing it as y = (x-5)/2, when x = 1, y = (1-5)/2 = -2, and when x = 5, y = (5-5)/2 = 0. So two solutions are (1, -2) and (5, 0).

    The full answer from Medha Ma’am →
14. Q14Long answer5 marksAsked: 2021

    Solve graphically : 9x - 10y + 12 = 0 and 2x + 3y - 13 = 0.

    How the model answer begins

    1. To represent the equation 9x - 10y + 12 = 0 graphically, we need at least two solutions of this equation. Writing the equation as y = (9x + 12)/10, we find two solutions. Putting x = 2, we get y = (18 + 12)/10 = 30/10 =…

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15. Q15Long answer5 marksAsked: 2021

    If 2 is added to both the numerator and denominator of a fraction, it becomes 1/2. If 7 is subtracted from both the numerator and denominator, it becomes 1/3. Find the fraction.

    How the model answer begins

    1. Let the numerator of the fraction be x and the denominator be y. Then the fraction is x/y.

    The full answer from Medha Ma’am →
16. Q16Short answer2 marksAsked: 2025

    The sum of two numbers is 50 and one number is 7/3 times of the other; then find the numbers.

    How the model answer begins

    1. Let the first number be x and the second number be y. Since the sum of the two numbers is 50, we get the equation x + y = 50.

    The full answer from Medha Ma’am →
17. Q17Short answer2 marksAsked: 2026

    Find the values of x and y of the pair of linear equations 3x + 2y = 12, 5x - 2y = 4 by cross-multiplication method.

    How the model answer begins

    1. The given pair of linear equations 3x + 2y = 12 and 5x - 2y = 4 is written in the standard form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, i.e. 3x + 2y - 12 = 0 and 5x - 2y - 4 = 0. Here a1 = 3, b1 = 2, c1 = -12, a2 =…

    The full answer from Medha Ma’am →
18. Q18Short answer2 marksAsked: 2026

    The perimeter of a right angled triangle is 40 cm. Its hypotenuse is 17 cm and the difference of the lengths of other two sides is 7 cm. Find the sides.

    How the model answer begins

    1. Let the two sides of the right-angled triangle other than the hypotenuse be x and y, where x is the longer side and y is the shorter side. Since the difference of the lengths of these two sides is 7 cm, we get x - y =…

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19. Q19Short answer2 marksAsked: 2026

    Find the value of k for which the pair of linear equations 3x + 5y = 0 and kx + 10y = 0 has a unique solution.

    How the model answer begins

    1. The pair of linear equations 3x + 5y = 0 and kx + 10y = 0 has a unique solution if the lines intersect in a single point, which happens when a1/a2 ≠ b1/b2. Here a1 = 3, b1 = 5, a2 = k, b2 = 10. So the condition for a…

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20. Q20Short answer2 marksAsked: 2026

    If the sum and difference of two numbers are 137 and 43 respectively then find the numbers.

    How the model answer begins

    1. Let the two numbers be x and y. According to the given condition, the sum of the numbers is 137, so the equation obtained is x + y = 137. The difference of the numbers is 43, so the equation obtained is x - y = 43.

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16 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   Graphical representation of the pair of linear equations x - y = 0, 3x + 4y - 20 = 0 is

   • (A) intersecting lines ✓
   • (B) coincident lines
   • (C) parallel lines
   • (D) none of these

   Answer: (A)

2. 2. Bihar Board 2026

   If a1/a2 = b1/b2 = c1/c2 in the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, then their solution is

   • (A) unique solution
   • (B) infinitely many solutions ✓
   • (C) no solution
   • (D) none of these

   Answer: (B)

3. 3. Bihar Board 2026

   If the graphical representation of pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 are parallel lines then

   • (A) a1/a2 != b1/b2
   • (B) a1/a2 = b1/b2 = c1/c2
   • (C) a1/a2 = b1/b2 != c1/c2 ✓
   • (D) none of these

   Answer: (C)

4. 4. Bihar Board 2026

   If the sum and difference of two numbers are 14 and 4 respectively then their product is

   • (A) 45 ✓
   • (B) 48
   • (C) 54
   • (D) 36

   Answer: (A)

5. 5. Bihar Board 2026

   If the speed of a boat in still water is 8 km/hr and the speed of the stream is 3 km/hr then relative speed of the boat in upstream is

   • (A) 11 km/hr
   • (B) 5 km/hr ✓
   • (C) 8 km/hr
   • (D) 3 km/hr

   Answer: (B)

6. 6. Bihar Board 2026

   A solution of linear equation x + 2y - 7 = 0 is

   • (A) x = 1, y = 5
   • (B) x = 2, y = 4
   • (C) x = 1, y = 3 ✓
   • (D) x = - 1, y = - 3

   Answer: (C)

7. 7. Bihar Board 2026

   How many solutions will 3x - 7y + 15 = 0, 6x - 14y - 32 = 0 have ?

   • (A) One solution
   • (B) No solution ✓
   • (C) Infinitely many solutions
   • (D) None of these

   Answer: (B)

8. 8. Bihar Board 2025

   Which of the following points lies on the graph of x = 2 ?

   • (A) (2, 0)
   • (B) (2, 1)
   • (C) (2, 2)
   • (D) all of these ✓

   Answer: (D)

9. 9. Bihar Board 2025

   How many solutions will x + 2y + 3 = 0, 3x + 6y + 9 = 0 have ?

   • (A) One solution
   • (B) No solution
   • (C) Infinitely many solutions ✓
   • (D) None of these

   Answer: (C)

10. 10. Bihar Board 2025

    If the graphs of two linear equations are parallel then the number of solutions will be

    • (A) 1
    • (B) 2
    • (C) infinitely many
    • (D) none of these ✓

    Answer: (D)

Questions parents and students ask

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

In the Bihar Board papers of 2021–2026, this chapter carried about 8 marks and about 8 questions a year; by marks it ranks 5 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 36 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.