Class 10 Mathematics: Pair of Linear Equations in Two Variables important questions

Pair of Linear Equations in Two Variables is a chapter of Class 10 Mathematics for the Andhra Pradesh State Board. These 17 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.

1. Practice · 1 marks

   1. "The cost of 2 pens and 5 pencils is ₹20." Express this as a linear equation.

   Let a pen cost ₹x and a pencil ₹y. Then 2x + 5y = 20.

2. Important · 2 marks

   2. If x + 2y = 7 and 4x − ay = 10 have no solution, find a.

   For no solution, a₁/a₂ = b₁/b₂ ≠ c₁/c₂. 1/4 = 2/(−a), so −a = 8 and a = −8. (Check: c₁/c₂ = 7/10 ≠ 1/4.)

3. Important · 4 marks

   3. Kavitha withdrew ₹8,000 from a bank and asked for ₹100 and ₹500 notes only. She got 32 notes in all. How many notes of each did she receive?

   Let her get x notes of ₹100 and y notes of ₹500. x + y = 32 and 100x + 500y = 8000, i.e. x + 5y = 80. Subtracting: 4y = 48, so y = 12 and x = 20. She received 20 notes of ₹100 and 12 notes of ₹500.

4. Practice · 1 marks

   4. Write the general form of a linear equation in two variables.

   ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero.

5. Very Important · 8 marks

   5. Solve graphically: 2x + y − 5 = 0 and 3x − 2y − 4 = 0.

   2x + y = 5: points (0, 5), (1, 3), (2, 1). 3x − 2y = 4: points (0, −2), (2, 1), (4, 4). The lines intersect at (2, 1). Solution: x = 2, y = 1.

6. Very Important · 8 marks

   6. 3 pens and 4 pencils together cost ₹44, whereas 4 pens and 3 pencils together cost ₹47. Form the pair of linear equations and solve graphically.

   Let a pen cost ₹x and a pencil ₹y. 3x + 4y = 44 and 4x + 3y = 47. Points for 3x + 4y = 44: (0, 11), (4, 8), (8, 5). Points for 4x + 3y = 47: (2, 13), (5, 9), (8, 5). The lines meet at (8, 5): a pen costs ₹8 and a pencil ₹5.

7. Practice · 1 marks

   7. Create a linear equation whose graph is parallel to the line 2x + 3y + 8 = 0.

   For example, 2x + 3y − 5 = 0. Parallel lines have a₁/a₂ = b₁/b₂ ≠ c₁/c₂, so keep the x and y coefficients (or the same ratio) and change the constant.

8. Very Important · 8 marks

   8. Draw the graphs of x − y + 1 = 0 and 3x + 2y − 12 = 0, and find the solution from the graph.

   x − y + 1 = 0, i.e. y = x + 1: points (0, 1), (1, 2), (2, 3). 3x + 2y − 12 = 0: points (0, 6), (2, 3), (4, 0). The lines intersect at (2, 3). Solution: x = 2, y = 3.

9. Very Important · 8 marks

   9. 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Form the pair of linear equations and find the cost of one pencil and one pen graphically.

   Let a pencil cost ₹x and a pen ₹y. 5x + 7y = 50 and 7x + 5y = 46. Points for 5x + 7y = 50: (3, 5), (10, 0), (−4, 10). Points for 7x + 5y = 46: (3, 5), (8, −2), (−2, 12). The lines meet at (3, 5). A pencil costs ₹3 and a pen costs ₹5.

10. Practice · 1 marks

    10. Write a pair of linear equations which are dependent.

    For example, x + y = 2 and 2x + 2y = 4. Here a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2, so the lines coincide and have infinitely many solutions.

11. Very Important · 8 marks

    11. Draw the graphs of 2x + 3y = 13 and 4x + 5y = 23 and find the solution from the graph.

    2x + 3y = 13: points (2, 3), (5, 1), (−1, 5). 4x + 5y = 23: points (2, 3), (−3, 7), (7, −1). The lines intersect at (2, 3). Solution: x = 2, y = 3.

12. Practice · 1 marks

    12. The pair of equations x − y = 1 and x + ky = 5 has a unique solution. If x = 2 and y = 1, find the value of k. A) 3 B) −3 C) 1/3 D) −1/3

    A) 3 Put x = 2, y = 1 in x + ky = 5: 2 + k = 5, so k = 3.

13. Important · 2 marks

    13. By comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether these equations are consistent or inconsistent: 5x − 3y = 11 −10x + 6y = 22

    a₁/a₂ = 5/(−10) = −1/2, b₁/b₂ = (−3)/6 = −1/2, c₁/c₂ = 11/22 = 1/2. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and have no solution. The pair is inconsistent.

14. Very Important · 8 marks

    14. Draw the graphs of the pair of linear equations x + 3y = 6 and 2x − 3y = 12, and find the solution from the graph.

    For x + 3y = 6: points (0, 2), (3, 1), (6, 0). For 2x − 3y = 12: points (0, −4), (3, −2), (6, 0). Plot both lines on the same axes. They intersect at (6, 0). Solution: x = 6, y = 0. (Check: 6 + 0 = 6 and 12 − 0 = 12.)

15. Practice · 1 marks

    15. Express y in terms of x from the equation 3x + 5y − 11 = 0.

    5y = 11 − 3x, so y = (11 − 3x)/5.

16. Important · 2 marks

    16. For what condition will 3x + ky = 7 and 6x + 4y = 14 have exactly one solution? Using the condition, find k.

    For exactly one solution, a₁/a₂ ≠ b₁/b₂. 3/6 ≠ k/4, so 1/2 ≠ k/4, which gives k ≠ 2. The pair has exactly one solution for every value of k except 2.

17. Very Important · 8 marks

    17. 10 students of Class X took part in a Mathematics quiz. The number of girls is 4 more than the number of boys. Form the pair of linear equations and find the number of boys and girls graphically.

    Let the number of boys be x and girls be y. x + y = 10 and y − x = 4. Points for x + y = 10: (0, 10), (5, 5), (10, 0). Points for y − x = 4: (0, 4), (2, 6), (3, 7). The lines intersect at (3, 7). So there are 3 boys and 7 girls.