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Linear Equations in Two Variables Class 9 Extra Questions | Mathematics Chapter 4

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Welcome to AssamSchool. This article contains 50 Extra Questions from Linear Equations in Two Variables from Class 9 Mathematics Chapter 4. These Linear Equations in Two Variables Class 9 Extra Questions are helpful to test your understanding and knowledge of the chapter. These additional questions will not only help you to excel in the chapter but also will give you a strong foundation in Mathematics.

We have also provided step-by-step answers to the questions for reference. So, let’s start…

Linear Equations in Two Variables Class 9 Extra Questions

50 Linear Equations in Two Variables Class 9 Extra Questions

1. The sum of two numbers is 50. Write a linear equation in two variables to represent this statement.

2. A pen costs ₹5 less than a pencil. Express this as a linear equation, taking the cost of a pen as \( x \) and a pencil as \( y \).

3. The perimeter of a rectangular garden is 30 meters. Formulate a linear equation if the length is \( l \) and the breadth is \( b \).

4. In a class, the number of girls is 10 more than twice the number of boys. Write an equation representing this situation.

5. A train travels at 60 km/h. Write an equation for the distance \( d \) covered in \( t \) hours. Is this a linear equation in two variables?

6. Convert \( 3x – 5y = 7.5 \) into the form \( ax + by + c = 0 \) and state \( a \), \( b\), \( c \).

7. Rewrite \( y = 4x – 2 \) in standard form and identify \( a \), \( b \), \( c \).

8. Express \( 0.5x + 2 = 1.3y \) in the form \( ax + by + c = 0 \).

9. Write \( \sqrt{2}x – \pi y = 4 \) in standard form.

10. Convert \( 7 – 3y = 2x \) into standard form.

11. For \( 4x – 3y + 6 = 0 \), what are \( a \), \( b \), and \( c \)?

12. Rewrite \( 5y = 10x + 3 \) in standard form and identify \( a \), \( b \), \( c \).

13. State \( a \), \( b \), \( c \) for \( 2.5x + \frac{7}{2}y – 3 = 0 \).

14. For \( -x + \frac{1}{3}y = 5 \), identify \( a \), \( b \), \( c \).

15. Is \( 0x + 0y = 5 \) a valid linear equation? Explain.

16. Find two solutions for \( x + y = 10 \).

17. Determine three solutions for \( 3x – 2y = 6 \).

18. If \( x = 4 \), find \( y \) in \( 2x + 5y = 18 \).

19. Choose \( y = -1 \) and find \( x \) in \( 4x – 3y = 11 \).

20. What is the solution of \( 0.x + 7y = 14 \) when \( x = 5 \)?

21. Check if \( (2, -3) \) is a solution of \( 5x + 2y = 4 \).

22. Verify if \( (0, 4) \) satisfies \( 3x – \frac{y}{2} = -2 \).

23. Is \( (-1, 2) \) a solution to \( 2x + 3y = 4 \)?

24. Does \( (3, -1) \) lie on the line \( 4x + 5y = 7 \)?

25. Which pairs are solutions to \( x – 3y = 6 \): \( (6,0)\), \( (0, -2) \), \( (3,1) \)?

26. Express \( x = -3 \) as a linear equation in two variables.

27. Write \( y = 7 \) in standard form and find two solutions.

28. The cost of 5 apples and 3 oranges is ₹45. Formulate a linear equation.

29. A car rental charges ₹200 base fee plus ₹10/hour. Write an equation for total cost \( C \) in terms of hours \( h \).

30. Does the point \( (2,0) \) lie on the line \( 2x + y = 4 \)?

31. Find \( k \) if the line \( kx + 3y = 6 \) passes through the point \( (2, -1) \).

32. Determine the condition for \( ax + by = c \) and \( dx + ey = f \) to represent parallel lines.

33. A linear equation has solutions \( (3, 4) \) and \( (5, 2) \). Derive the equation.

34. Write the equation of the line passing through \( (1, 2) \) and \( (3, 4) \) in standard form.

35. For what value of \( m \) do \( mx – 5y = 7 \) and \( 2x + y = 3 \) have no solution?

36. Calculate the area of the triangle formed by \( x=0 \), \( y=0 \), and \( 3x + 4y = 12 \).

37. If \( (k, 2k) \) lies on \( 2x – 3y = 5 \), find \( k \).

38. A boat travels 30 km upstream and 44 km downstream in 10 hours. It also travels 40 km upstream and 55 km downstream in 13 hours. Formulate equations for the boat’s speed in still water and the stream’s speed.

39. Solve for \( x \) and \( y \): \( \frac{x}{2} + \frac{3y}{2} = 6 \) and \( x – y = 3 \).

40. Prove that the lines \( 4x – 6y = 8 \) and \( 2x – 3y = 4 \) are coincident.

41. Find \( p \) if \( px + 2y = 5 \) and \( 3x + y = 1 \) have infinitely many solutions.

42. A rectangle’s length is 5 more than its width. If the perimeter is 50, find the dimensions.

43. If \( (2, 3) \) satisfies both \( 3x + ay = 12 \) and \( ax + y = 7 \), find \( a \).

44. A father is three times as old as his son. In 10 years, he will be twice as old. Find their current ages.

45. Find the equation of the line perpendicular to \( 2x + 3y = 6 \) and passing through \( (1, 1) \).

46. A chemist mixes a 20% acid solution with a 50% acid solution to get 12 liters of 40% acid. Formulate equations.

47. The cost of 3 pens and 4 pencils is ₹50. A pen costs twice a pencil. Find the cost of each.

48. Solve for \( x \): \( \frac{2x + y}{3} = 5 \) and \( x – \frac{y}{2} = 1 \).

49. Find the intersection points of \( 2x + y = 8 \) with the x-axis and y-axis, then compute the distance between them.

50. Show that \( y = 2x + 1 \) and \( 4x – 2y + 3 = 0 \) are parallel.

Solutions

  1. Answer:
    Let the two numbers be \( x \) and \( y \)
    \( x + y = 50 \)
  2. Answer:
    \( x = y – 5 \) or \( x – y + 5 = 0 \)
  3. Answer:
    Perimeter \( = 2(l + b) = 30 \)
    \( 2l + 2b = 30 \) or \( l + b = 15 \)
  4. Answer:
    Let the number of boys be \( b \), girls be \( g \)
    \( g = 2b + 10 \) or \( g – 2b – 10 = 0 \)
  5. Answer:
    \( d = 60t \). Yes, it is a linear equation in two variables \(( d )\) and \(( t )\)
  6. Answer:
    \( 3x – 5y – 7.5 = 0 \)
    \( a = 3 \), \( b = -5 \), \( c = -7.5 \)
  7. Answer:
    \( 4x – y – 2 = 0 \)
    \( a = 4 \), \( b = -1 \), \( c = -2 \)
  8. Answer:
    \( 0.5x – 1.3y + 2 = 0 \)
    Multiply by 10 to eliminate decimals: \( 5x – 13y + 20 = 0 \)
  9. Answer:
    \( \sqrt{2}x – \pi y – 4 = 0 \)
    \( a = \sqrt{2} \), \( b = -\pi \), \( c = -4 \)
  10. Answer:
    \( 2x + 3y – 7 = 0 \)
  11. Answer:
    \( a = 4 \), \( b = -3 \), \( c = 6 \)
  12. Answer:
    \( 10x – 5y + 3 = 0 \)
    \( a = 10 \), \( b = -5 \), \( c = 3 \)
  13. Answer:
    \( a = 2.5 \), \( b = \frac{7}{2} \) or \(3.5\), \( c = -3 \)
  14. Answer:
    \( a = -1 \), \( b = \frac{1}{3} \), \( c = -5 \)
  15. Answer:
    No. A linear equation requires \( a \) and \( b \) not both zero. Here, \( a = 0 \), \( b = 0 \), which is invalid.
  16. Answer:
    • Let \( x = 0 \): \( y = 10 \) → \( (0, 10) \)
    • Let \( y = 0 \): \( x = 10 \) → \( (10, 0) \)
  17. Answer:
    • \( x = 0 \): \( -2y = 6\) → \( y = -3 \) → \( (0, -3) \)
    • \( y = 0 \): \( 3x = 6 \) → \( x = 2 \) → \( (2, 0) \)
    • \( x = 4 \): \( 12 – 2y = 6 \) → \( y = 3 \) → \( (4, 3) \)
  18. Answer:
    Substitute \( x = 4 \):
    \( 2(4) + 5y = 18 \) → \( 8 + 5y = 18 \) → \( y = 2 \)
  19. Answer:
    Substitute \( y = -1 \):
    \( 4x – 3(-1) = 11 \) → \( 4x + 3 = 11 \) → \( x = 2 \)
  20. Answer:
    \( 7y = 14 \) → \( y = 2 \). For any \( x \), \( y = 2 \). Solution: \( (5, 2) \)
  21. Answer:
    Substitute \( (2, -3) \):
    \( 5(2) + 2(-3) = 10 – 6 = 4 \). Yes, it is a solution.
  22. Answer:
    Substitute \( (0, 4) \):
    \( 3(0) – \frac{4}{2} = -2 ). ( 0 – 2 = -2 \). Yes, it satisfies.
  23. Answer:
    Substitute \( (-1, 2) \):
    \( 2(-1) + 3(2) = -2 + 6 = 4 \). Yes, it is a solution.
  24. Answer:
    Substitute \( (3, -1) \):
    \( 4(3) + 5(-1) = 12 – 5 = 7 \). Yes, it lies on the line
  25. Answer:
    • \( (6,0) \): \( 6 – 3(0) = 6 \) → Yes
    • \( (0, -2) \): \( 0 – 3(-2) = 6 \) → Yes
    • \( (3,1) \): \( 3 – 3(1) = 0 \neq 6 \) → No
  26. Answer:
    \( 1x + 0y + 3 = 0 \)
  27. Answer:
    Standard form: \( 0x + 1y – 7 = 0 \)
    Solutions: \( (0, 7) ), ( (5, 7) \)
  28. Answer:
    Let cost of an apple = \( a \), orange = \( o \)
    \( 5a + 3o = 45 \)
  29. Answer:
    \( C = 10h + 200 \)
  30. Answer:
    Substitute \( (2,0) \): \( 2(2) + 0 = 4 \). Yes, it lies on the line.
  31. Answer:
    Substitute \( (2, -1) \) into \( kx + 3y = 6 \):
    \( k(2) + 3(-1) = 6 \) → \( 2k – 3 = 6 \) → \( 2k = 9 \) → \( k = 4.5 \)
  32. Answer:
    For parallel lines, their slopes must be equal.
    Rewrite equations as \( y = -\frac{a}{b}x + \frac{c}{b} \) and \( y = -\frac{d}{e}x + \frac{f}{e} \)
    Condition: \( \frac{a}{b} = \frac{d}{e} \) or \( ae = bd \)
  33. Answer:
    Let the equation be \( ax + by = c \)
    Substitute \( (3, 4) \): \( 3a + 4b = c \)
    Substitute \( (5, 2) \): \( 5a + 2b = c \)
    Subtract: \( 2a – 2b = 0 \) → \( a = b \)
    Let \( a = b = 1 \), then \( 3(1) + 4(1) = 7 = c \)
    Equation: \( x + y = 7 \)
  34. Answer:
    Slope \( m = \frac{4-2}{3-1} = 1 \)
    Equation: \( y – 2 = 1(x – 1) \) → \( y = x + 1 \)
    Standard form: \( x – y + 1 = 0 \)
  35. Answer:
    For no solution, the lines must be parallel:
    \( \frac{m}{2} = \frac{-5}{1} \neq \frac{7}{3} \)
    Thus, \( m = -10 \)
  36. Answer:
    X-intercept: \( x = 4 \), Y-intercept: \( y = 3 \)
    Area = \( \frac{1}{2} \times 4 \times 3 = 6 \) sq. units
  37. Answer:
    Substitute \( (k, 2k) \):
    \( 2k – 3(2k) = 5 \) → \( 2k – 6k = 5 \) → \( -4k = 5 \) → \( k = -\frac{5}{4} \)
  38. Answer:
    Let boat speed = \( b \), stream speed = \( s \)
    Upstream: \( \frac{30}{b – s} + \frac{44}{b + s} = 10 \)
    Downstream: \( \frac{40}{b – s} + \frac{55}{b + s} = 13 \)
  39. Answer:
    Multiply first equation by 2: \( x + 3y = 12 \)
    Subtract second equation \( x – y = 3 \):
    \( 4y = 9 \) → \( y = 2.25 \), then \( x = 5.25 \)
  40. Answer:
    Divide \( 4x – 6y = 8 \) by 2: \( 2x – 3y = 4 \), which matches the second equation. Hence, coincident.
  41. Answer:
    For infinitely many solutions, ratios must be equal:
    \( \frac{p}{3} = \frac{2}{1} = \frac{5}{1} \)
    \( p = 6 \)
  42. Answer:
    Let width = \( w \), length = \( w + 5 \)
    Perimeter: \( 2(w + w + 5) = 50 \) → \( 2w + 5 = 25 \) → \( w = 10 \)
    Dimensions: \( 10 \times 15 \)
  43. Answer:
    Substitute \( (2, 3) \):
    \( 3(2) + a(3) = 12 \) → \( 6 + 3a = 12 \) → \( a = 2 \)
    Verify in \( 2x + y = 7 \): \( 2(2) + 3 = 7 \) → Valid
  44. Answer:
    Let son’s age = \( s \), father’s age = \( 3s \)
    In 10 years: \( 3s + 10 = 2(s + 10) \) → \( 3s + 10 = 2s + 20 \) → \( s = 10 \)
    Father: 30, Son: 10
  45. Answer:
    Slope of \( 2x + 3y = 6 \) is \( -\frac{2}{3} \)
    Perpendicular slope = \( \frac{3}{2} \)
    Equation: \( y – 1 = \frac{3}{2}(x – 1) \) → \( 3x – 2y = 1 \)
  46. Answer:
    Let 20% solution = \( x \) L, 50% solution = \( y \) L
    \( x + y = 12 \),
    \( 0.2x + 0.5y = 0.4(12) \)
  47. Solution:
    Let pencil cost = \( p \), pen cost = \( 2p \)
    \( 3(2p) + 4p = 50 \) → \( 10p = 50 \) → \( p = 5 \)
    Pen: ₹10, Pencil: ₹5
  48. Answer:
    Multiply second equation by 3: \( 3x – \frac{3y}{2} = 3 \)
    Add to first equation: \( \frac{2x + y}{3} + 3x – \frac{3y}{2} = 8 \)
    Solve to find \( x = 3 \), \( y = 6 \)
  49. Answer:
    X-intercept: \( y = 0 \) → \( x = 4 \)
    Y-intercept: \( x = 0 \) → \( y = 8 \)
    Distance = \( \sqrt{4^2 + 8^2} = \sqrt{80} = 4\sqrt{5} \)
  50. Answer:
    Rewrite \( 4x – 2y + 3 = 0 \) as \( y = 2x + 1.5 \)
    Both lines have slope 2 but different y-intercepts. Hence, parallel.

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The Bottom Line

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You can also check the related MCQs for this chapter here: MCQ on Linear Equations in Two Variables

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