Loading...

About Simultaneous Linear Equations — Class 9 ICSE

Solve pairs of linear equations using substitution, elimination, and cross-multiplication methods. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 5). On this page you can practice 59 questions across three difficulty levels — 20 easy, 20 medium, and 19 hard — each with a visual step-by-step solution, plus a timed 34-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Simultaneous Linear Equations

  • Introduction to Simultaneous Linear Equations
  • Method 1: Substitution Method
  • Method 2: Elimination Method
  • Method 3: Cross-Multiplication Method
  • Summary, Connections, and Practice

Interactive lesson · about 15 minutes · checkpoint question after every unit

Simultaneous Linear Equations — solved examples for Class 9 ICSE

Example 1easy

Which of the following equations is a linear equation in two variables, x and y?
  1. A)x² + y = 5
  2. B)3x - 4y = xy
  3. C)2x + 5y = 10
  4. D)√x + y = 7

Step-by-step solution

  1. A linear equation in two variables (x and y) has the general form Ax + By + C = 0, where A, B, and C are real numbers, and A and B are not both zero.
  2. In such an equation, the highest power of each variable is 1, and there are no product terms like xy.
  3. Option A has x², B has xy, and D has √x (x to the power of 1/2). Option C, 2x + 5y = 10, fits this definition.

Answer: 2x + 5y = 10

Example 2medium

Solve the system of equations using the substitution method: 3x - y = 7 and 2x + 3y = 1.
  1. A)x=2, y=-1
  2. B)x=1, y=-4
  3. C)x=3, y=2
  4. D)x=-1, y=-10

Step-by-step solution

  1. From the first equation, 3x - y = 7, we can express y as: y = 3x - 7.
  2. Substitute this expression for y into the second equation: 2x + 3(3x - 7) = 1.
  3. Simplify and solve for x: 2x + 9x - 21 = 1 => 11x = 22 => x = 2.
  4. Substitute x=2 back into y = 3x - 7: y = 3(2) - 7 = 6 - 7 = -1. So, the solution is x=2, y=-1.

Answer: x=2, y=-1

Example 3hard

Consider the system of equations: (x+y)/2 + (x-y)/3 = 7 and (x+y)/3 + (x-y)/4 = 5. What is the value of x?
  1. A)5
  2. B)7
  3. C)9
  4. D)11

Step-by-step solution

  1. Let u = x+y and v = x-y. The given equations become:
  2. Equation 1: u/2 + v/3 = 7 => 3u + 2v = 42 (multiplying by 6)
  3. Equation 2: u/3 + v/4 = 5 => 4u + 3v = 60 (multiplying by 12)
  4. Now, solve the system: (1) 3u + 2v = 42; (2) 4u + 3v = 60.
  5. Multiply (1) by 3: 9u + 6v = 126. Multiply (2) by 2: 8u + 6v = 120. Subtract the second new equation from the first: (9u - 8u) + (6v - 6v) = 126 - 120 => u = 6.
  6. Substitute u = 6 into (1): 3(6) + 2v = 42 => 18 + 2v = 42 => 2v = 24 => v = 12.
  7. We have u = x+y = 6 and v = x-y = 12. Add these two equations: (x+y) + (x-y) = 6 + 12 => 2x = 18 => x = 9.

Answer: 9

Practice questions on Simultaneous Linear Equations

  1. Q1.easy

    When solving a pair of simultaneous linear equations graphically, what does the point where the two lines intersect represent?
    1. A)A point that satisfies only the first equation.
    2. B)The unique solution that satisfies both equations simultaneously.
    3. C)A point that satisfies only the second equation.
    4. D)The origin of the graph.
    Show answer

    Answer: The unique solution that satisfies both equations simultaneously.

    Hint: Each point on a line represents a solution to its equation. What happens at the intersection?

  2. Q2.easy

    Rohan was solving the system of equations: (1) x + 2y = 7 and (2) 3x - y = 1. He decided to use the substitution method. His first step was to express x from equation (1) as x = 7 - 2y and substitute it into equation (2). Which of the following is the *correct* next step?
    1. A)3(7) - y = 1
    2. B)3(7 - 2y) - y = 1
    3. C)(7 - 2y) + 2y = 7
    4. D)x + 2(7 - 2y) = 7
    Show answer

    Answer: 3(7 - 2y) - y = 1

    Hint: When you substitute an expression for a variable, you must replace *every* instance of that variable in the *other* equation.

  3. Q3.easy

    Consider the system of equations: (1) 4x + 3y = 10 and (2) 2x - 5y = -8. To solve this system using the elimination method, which of the following initial steps would be the *most efficient* to eliminate x?
    1. A)Multiply equation (2) by 2.
    2. B)Multiply equation (1) by -1.
    3. C)Multiply equation (1) by 5 and equation (2) by 3.
    4. D)Add equation (1) and equation (2).
    Show answer

    Answer: Multiply equation (2) by 2.

    Hint: To eliminate x, you need its coefficients to be additive inverses (e.g., 4x and -4x). How can you easily make 2x become 4x or -4x?

  4. Q4.medium

    Solve the system of equations using the elimination method: 5x + 2y = 16 and 3x - 4y = 7.
    1. A)x=2, y=3
    2. B)x=3, y=1/2
    3. C)x=4, y=-2
    4. D)x=1, y=11/2
    Show answer

    Answer: x=3, y=1/2

    Hint: Multiply one or both equations by suitable numbers to make the coefficients of one variable equal or additive inverses, then add or subtract the equations.

  5. Q5.medium

    The sum of two numbers is 50. If the larger number is decreased by 10 and the smaller number is increased by 5, the new larger number is twice the new smaller number. Find the original numbers.
    1. A)35, 15
    2. B)25, 25
    3. C)40, 10
    4. D)30, 20
    Show answer

    Answer: 40, 10

    Hint: Let the two numbers be x and y. Form two equations based on the given conditions. Pay attention to how the numbers change for the second condition.

  6. Q6.medium

    Five years ago, a father was three times as old as his son. Ten years later, the father will be twice as old as his son. What are their present ages?
    1. A)Father = 55, Son = 25
    2. B)Father = 50, Son = 20
    3. C)Father = 45, Son = 15
    4. D)Father = 60, Son = 30
    Show answer

    Answer: Father = 50, Son = 20

    Hint: Define variables for their present ages. Create expressions for their ages five years ago and ten years later, then form the equations.

  7. Q7.hard

    The sum of a two-digit number and the number formed by reversing its digits is 110. If 10 is subtracted from the original number, the new number is 4 more than 5 times the sum of its digits. Find the original two-digit number.
    1. A)46
    2. B)37
    3. C)64
    4. D)73
    Show answer

    Answer: 37

    Hint: Represent the two-digit number as 10x + y, where x is the tens digit and y is the units digit. Formulate two linear equations based on the given conditions.

  8. Q8.hard

    If the numerator of a fraction is increased by 2 and the denominator is decreased by 1, the fraction becomes 3/2. If the numerator is decreased by 4 and the denominator is increased by 2, the fraction becomes 1/3. Find the original fraction.
    1. A)7/5
    2. B)5/7
    3. C)9/7
    4. D)7/9
    Show answer

    Answer: 7/5

    Hint: Let the fraction be x/y. Formulate two linear equations by cross-multiplying after applying the given conditions to the numerator and denominator.

  9. Q9.hard

    For what value of 'k' will the system of linear equations 3x + ky = 7 and 6x + 4y = 10 represent parallel lines?
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 2

    Hint: For two linear equations to represent parallel lines (no solution), the ratio of their x-coefficients must be equal to the ratio of their y-coefficients, but not equal to the ratio of their constant terms.

These are 9 of the 59 questions available for Simultaneous Linear Equations. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.