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About Pair of Linear Equations in Two Variables — Class 10 CBSE

Solve pairs of linear equations graphically and algebraically using substitution and elimination. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 3). On this page you can practice 67 questions across three difficulty levels — 26 easy, 21 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 36-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Pair of Linear Equations in Two Variables

  • Introduction to Pair of Linear Equations in Two Variables
  • Graphical Method for Solving Pairs of Linear Equations
  • Algebraic Methods: Substitution and Elimination
  • Algebraic Method: Cross-Multiplication & Advanced Cases
  • Summary, Connections, and Practice Preparation

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

Pair of Linear Equations in Two Variables — solved examples for Class 10 CBSE

Example 1easy

For a pair of linear equations in two variables, (x, y) = (p, q) is a solution if:
  1. A)It satisfies the first equation only.
  2. B)It satisfies the second equation only.
  3. C)It satisfies both equations simultaneously.
  4. D)It is a point on the x-axis.

Step-by-step solution

  1. A solution to a pair of linear equations must satisfy both equations when the values of x and y are substituted into them.
  2. Graphically, this solution represents the point where the two lines corresponding to the equations intersect.

Answer: It satisfies both equations simultaneously.

Example 2medium

For which value of 'k' will the pair of linear equations 3x + y = 1 and (2k - 1)x + (k - 1)y = 2k + 1 have no solution?
  1. A)k = 1
  2. B)k = 2
  3. C)k = -1
  4. D)k = 0

Step-by-step solution

  1. The given equations are: 3x + y - 1 = 0 and (2k - 1)x + (k - 1)y - (2k + 1) = 0.
  2. For no solution, we must have a₁/a₂ = b₁/b₂ ≠ c₁/c₂. So, 3/(2k - 1) = 1/(k - 1).
  3. Cross-multiplying, we get 3(k - 1) = 1(2k - 1), which simplifies to 3k - 3 = 2k - 1.
  4. Solving for k: 3k - 2k = -1 + 3, so k = 2. We also need to check that 1/(k-1) ≠ -1/-(2k+1) for k=2. 1/(2-1) = 1/1 = 1. And (2k+1)/1 = (2(2)+1)/1 = 5. So 1 ≠ 5, which satisfies the condition.

Answer: k = 2

Example 3hard

For what values of 'k' and 'm' does the following system of linear equations have infinitely many solutions?
(k-3)x + 3y = k
kx + ky = 12
  1. A)k=6, m is arbitrary
  2. B)k=6, m=2
  3. C)k=3, m=4
  4. D)k=6, m=6

Step-by-step solution

  1. For infinitely many solutions, the ratio of coefficients must be equal: a1/a2 = b1/b2 = c1/c2.
  2. From the given equations, a1 = k-3, b1 = 3, c1 = k; and a2 = k, b2 = k, c2 = 12.
  3. Set up the ratios: (k-3)/k = 3/k = k/12.
  4. From 3/k = k/12, we get k² = 36, so k = ±6. Since (k-3)/k is involved, let's check k=6 and k=-6. If k=6, then (6-3)/6 = 3/6 = 1/2, and 3/6 = 1/2, and 6/12 = 1/2. All ratios are equal. If k=-6, then (-6-3)/(-6) = -9/-6 = 3/2, and 3/(-6) = -1/2. These are not equal, so k=-6 is not a solution. Thus, k=6.
  5. The variable 'm' is not present in the given system of equations, hence it is not constrained by these equations. Therefore, k=6 and 'm' can be any real number (arbitrary).

Answer: k=6, m is arbitrary

Practice questions on Pair of Linear Equations in Two Variables

  1. Q1.easy

    If the graph of a pair of linear equations in two variables shows two lines intersecting at a single point, then the system of equations has:
    1. A)No solution.
    2. B)Exactly one solution.
    3. C)Infinitely many solutions.
    4. D)Two solutions.
    Show answer

    Answer: Exactly one solution.

    Hint: The solution to a system of equations is represented by the point(s) where their graphs meet.

  2. Q2.easy

    For what value of 'k' will the pair of linear equations x + ky = 3 and 3x + 2y = 1 NOT have a unique solution?
    1. A)2/3
    2. B)1/3
    3. C)-2/3
    4. D)-1/3
    Show answer

    Answer: 2/3

    Hint: For a system to NOT have a unique solution, the ratio a1/a2 must be equal to b1/b2.

  3. Q3.easy

    Ravi is trying to solve the system: x + y = 7 and 2x - 3y = 4 using the substitution method. He correctly expresses x from the first equation as x = 7 - y. Which of the following is the correct next step to substitute into the second equation?
    1. A)2(7 - y) - 3y = 4
    2. B)7 - y - 3y = 4
    3. C)2x - 3(7 - y) = 4
    4. D)2(7 - y) - 3(7 - y) = 4
    Show answer

    Answer: 2(7 - y) - 3y = 4

    Hint: The expression for 'x' should replace 'x' in the *other* equation, while the 'y' term in that equation remains as is.

  4. Q4.medium

    The cost of 5 pens and 7 notebooks is ₹255, while the cost of 7 pens and 5 notebooks is ₹249. What is the cost of one pen and one notebook, respectively?
    1. A)₹20, ₹25
    2. B)₹22, ₹21
    3. C)₹25, ₹20
    4. D)₹21, ₹22
    Show answer

    Answer: ₹22, ₹21

    Hint: Set up a pair of linear equations representing the given information. Let 'x' be the cost of a pen and 'y' be the cost of a notebook, then solve the system.

  5. Q5.medium

    The cost of 5 pens and 7 notebooks is ₹257, while the cost of 7 pens and 5 notebooks is ₹259. What is the cost of one pen and one notebook, respectively?
    1. A)₹20, ₹25
    2. B)₹22, ₹21
    3. C)₹25, ₹20
    4. D)₹21, ₹22
    Show answer

    Answer: ₹22, ₹21

    Hint: Set up a pair of linear equations representing the given information. Let 'x' be the cost of a pen and 'y' be the cost of a notebook, then solve the system.

  6. Q6.medium

    Solve the following pair of linear equations using the elimination method: x + 2y = 7 and 2x - y = 4.
    1. A)x = 3, y = 2
    2. B)x = 2, y = 3
    3. C)x = 1, y = 3
    4. D)x = 3, y = 1
    Show answer

    Answer: x = 3, y = 2

    Hint: Multiply one or both equations by suitable numbers so that the coefficients of one variable become equal in magnitude and opposite in sign. Then add the equations.

  7. Q7.hard

    A father's age is three times the sum of the ages of his two children. After 5 years, his age will be two times the sum of their ages. Find the father's current age.
    1. A)45 years
    2. B)50 years
    3. C)55 years
    4. D)60 years
    Show answer

    Answer: 45 years

    Hint: Let the father's current age be 'F' and the sum of the children's current ages be 'S'. Form two equations based on the given information, remembering that after 5 years, the sum of the children's ages will increase by 5 + 5 = 10 years.

  8. Q8.hard

    Consider the pair of linear equations: L1: (k+1)x + 8y = 4k and L2: kx + (k+3)y = 3k-1. If these lines are parallel, which of the following statements must be true?
    1. A)k = -3
    2. B)k = 1
    3. C)k = 3
    4. D)k = -1
    Show answer

    Answer: k = 1

    Hint: For two lines to be parallel, 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. Focus on the first equality to find 'k'.

  9. Q9.hard

    Solve the following pair of linear equations for x and y:
    (a-b)x + (a+b)y = a² - 2ab - b²
    (a+b)x + (a+b)y = a² + b²
    1. A)x = b, y = -a
    2. B)x = a-b, y = (a² + b²)/(a+b)
    3. C)x = a, y = -b
    4. D)x = a+b, y = (a² - 2ab - b²)/(a+b)
    Show answer

    Answer: x = a-b, y = (a² + b²)/(a+b)

    Hint: Notice that the 'y' coefficients in the second equation and the first equation are quite similar. Try subtracting the second equation from the first to eliminate 'y' and find 'x'.

These are 9 of the 67 questions available for Pair of Linear Equations in Two Variables. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.