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About Linear Equations in Two Variables — Class 9 Olympiad

Solve and graph linear equations in two variables; find integer solutions and apply to word problems. This topic is part of the Olympiad Class 9 mathematics syllabus (chapter: Module 4). On this page you can practice 49 questions across three difficulty levels — 19 easy, 10 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Linear Equations in Two Variables — solved examples for Class 9 Olympiad

Example 1easy

The ordered pair (p, q) is a solution to the linear equation 3x - 5y = 15. What is the value of 6p - 10q?
  1. A)15
  2. B)20
  3. C)30
  4. D)45

Step-by-step solution

  1. Given that (p, q) is a solution to 3x - 5y = 15, we can substitute x = p and y = q into the equation.
  2. So, we have 3p - 5q = 15.
  3. We need to find the value of 6p - 10q. Notice that 6p - 10q is exactly twice the expression 3p - 5q.
  4. Multiply both sides of the equation 3p - 5q = 15 by 2: 2 × (3p - 5q) = 2 × 15, which gives 6p - 10q = 30.

Answer: 30

Example 2medium

Given the linear equation ax + by + c = 0. If (1, 2) is a solution and a - b + c = 0, then which of the following points must also be a solution to the equation?
  1. A)(0, 0)
  2. B)(2, 1)
  3. C)(1, -5)
  4. D)(5, 1)

Step-by-step solution

  1. Since (1, 2) is a solution to ax + by + c = 0, we have a(1) + b(2) + c = 0, which simplifies to a + 2b + c = 0 (Equation 1).
  2. We are also given the condition a - b + c = 0 (Equation 2).
  3. Subtract Equation 2 from Equation 1: (a + 2b + c) - (a - b + c) = 0 - 0, which simplifies to 3b = 0, so b = 0.
  4. Substitute b = 0 into Equation 1: a + 2(0) + c = 0, which means a + c = 0, or c = -a. Therefore, the linear equation becomes ax + 0y - a = 0, or ax - a = 0. Assuming a is not 0 (otherwise it's a degenerate equation 0=0), we can divide by a to get x - 1 = 0, or x = 1. The line is a vertical line at x = 1. Any point with an x-coordinate of 1 must be a solution. Among the given options, (1, -5) has an x-coordinate of 1.
  5. Thus, the point (1, -5) must also be a solution to the equation.

Answer: (1, -5)

Example 3hard

How many pairs of positive integers (x, y) satisfy the equation 7x + 11y = 200, such that x < y?
  1. A)1
  2. B)2
  3. C)3
  4. D)4

Step-by-step solution

  1. We need to find positive integer solutions (x, y) for 7x + 11y = 200 with x < y.
  2. By inspection or using modular arithmetic, we can find a particular solution. For example, if y=15, 7x + 11(15) = 200 => 7x + 165 = 200 => 7x = 35 => x = 5. So (5, 15) is a solution.
  3. The general solution for 7x + 11y = 200 is given by x = 5 + 11k and y = 15 - 7k for some integer k.
  4. Applying the conditions: (1) x > 0 => 5 + 11k > 0 => 11k > -5 => k >= 0. (2) y > 0 => 15 - 7k > 0 => 15 > 7k => k <= 2. Possible integer k values are 0, 1, 2. (3) x < y => 5 + 11k < 15 - 7k => 18k < 10 => k < 10/18 => k < 5/9. So k must be 0.
  5. For k=0, (x, y) = (5, 15). This satisfies x > 0, y > 0, and x < y (5 < 15).
  6. For k=1, (x, y) = (16, 8). This does not satisfy x < y (16 is not less than 8).
  7. For k=2, (x, y) = (27, 1). This does not satisfy x < y (27 is not less than 1).
  8. Thus, there is only 1 such pair of positive integers.

Answer: 1

Practice questions on Linear Equations in Two Variables

  1. Q1.easy

    In a school, the number of teachers (t) is 5 less than three times the number of classrooms (c). Which of the following linear equations correctly represents this relationship?
    1. A)3t - c = 5
    2. B)t = 3c - 5
    3. C)c = 3t - 5
    4. D)t + c = 5
    Show answer

    Answer: t = 3c - 5

    Hint: Translate the phrase 'is 5 less than three times' carefully into an algebraic expression. Remember which variable is dependent on the other.

  2. Q2.easy

    A line passes through the point (-2, 4) and is parallel to the x-axis. Which of the following points must also lie on this line?
    1. A)(4, -2)
    2. B)(-2, 0)
    3. C)(0, 4)
    4. D)(4, 4)
    Show answer

    Answer: (0, 4)

    Hint: Recall that a line parallel to the x-axis has a constant y-coordinate for all its points.

  3. Q3.easy

    The point (a, a + 2) is a solution to the equation 5x - 3y = 4. What is the value of 'a'?
    1. A)-5
    2. B)-1
    3. C)1
    4. D)5
    Show answer

    Answer: 5

    Hint: Substitute the given coordinates (x = a, y = a + 2) into the linear equation and solve for 'a'.

  4. Q4.medium

    Find the number of positive integer solutions (x, y) to the equation 5x + 3y = 40 such that x is an odd number.
    1. A)0
    2. B)2
    3. C)1
    4. D)3
    Show answer

    Answer: 1

    Hint: First, find the range of possible integer values for y. Then, test which of these values lead to integer solutions for x, keeping in mind the additional condition that x must be odd.

  5. Q5.medium

    The line ax + by = 6 passes through the point (0, 3) and has an x-intercept of (-2, 0). What is the area of the triangle formed by this line and the coordinate axes?
    1. A)6 sq. units
    2. B)9 sq. units
    3. C)12 sq. units
    4. D)3 sq. units
    Show answer

    Answer: 3 sq. units

    Hint: Use the given intercepts to determine the equation of the line. The intercepts directly provide the base and height of the triangle formed with the axes.

  6. Q6.medium

    A square ABCD has two of its vertices, A and B, on the x-axis. The vertex C has coordinates (5, 3). Which of the following equations represents the line passing through D and parallel to the y-axis?
    1. A)y = 3
    2. B)x = 2
    3. C)x = 5
    4. D)y = 2
    Show answer

    Answer: x = 2

    Hint: Utilize the properties of a square and the given coordinates to find the coordinates of vertex D. A line parallel to the y-axis has the form x = constant.

  7. Q7.hard

    The graph of the equation ax + by = 1 passes through the points (2, -1) and (-3, 4). What is the area of the triangle formed by this line and the coordinate axes?
    1. A)1/4 square units
    2. B)1/2 square units
    3. C)1 square unit
    4. D)2 square units
    Show answer

    Answer: 1/2 square units

    Hint: First, use the given points to find the coefficients 'a' and 'b' of the linear equation. Then, determine the x-intercept and y-intercept to calculate the triangle's area.

  8. Q8.hard

    A number consists of two digits. The sum of the digits is 12. If 18 is subtracted from the number, the digits are reversed. If the digits are reversed, and then the new number is divided by the original number, what is the remainder?
    1. A)5
    2. B)12
    3. C)57
    4. D)75
    Show answer

    Answer: 57

    Hint: Form two linear equations based on the conditions for the two-digit number and its reversed form. Solve for the digits, then perform the final division.

  9. Q9.hard

    For how many integer pairs (x, y) does the equation |x| + |y| = 5 hold?
    1. A)10
    2. B)16
    3. C)18
    4. D)20
    Show answer

    Answer: 20

    Hint: Consider the symmetry of the absolute value function. Analyze each quadrant and the axes separately, remembering that x and y must be integers.

These are 9 of the 49 questions available for Linear Equations in Two Variables. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.