Pair of Linear Equations in Two Variables Important Questions.

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1. Q1Easy

   Consider the pair of linear equations: x + 3y = 6 and 2x – 3y = 12. What does the graphical representation indicate about their consistency?

   • A.The equations are inconsistent and have no solution.
   • B.The equations are consistent and have infinitely many solutions.
   • C.The equations are consistent and have a unique solution.
   • D.The equations are dependent and represent the same line.
   Answer: The equations are consistent and have a unique solution.

   Explanation: The lines intersect at a single point, confirming a unique solution exists.

2. Q2Easy

   Which method is used to solve the pair of equations: x + y = 5 and 2x – y = 1?

   • A.Substitution Method
   • B.Graphical Method
   • C.Elimination Method
   • D.Consistency Check
   Answer: Elimination Method

   Explanation: The elimination method is used by adding the equations to eliminate y and solve for x.

3. Q3Easy

   What is the graphical representation of the pair of equations: x + y = 3 and x + y = 5?

   • A.Intersecting lines
   • B.Coincident lines
   • C.Parallel lines
   • D.Perpendicular lines
   Answer: Parallel lines

   Explanation: The equations have the same slope but different y-intercepts, indicating parallel lines.

4. Q4Easy

   For the pair of equations x + y = 5 and 2x – 3y = 4, which method is most convenient to solve them?

   • A.Substitution Method
   • B.Elimination Method
   • C.Graphical Method
   • D.Both Substitution and Elimination Methods are equally convenient.
   Answer: Elimination Method

   Explanation: The elimination method is straightforward here as the coefficients of y can be easily aligned for elimination.

5. Q5Medium

   Consider the pair of linear equations: x + 3y = 6 and 2x – 3y = 12. What is the graphical representation of these equations?

   • A.The lines are parallel and distinct.
   • B.The lines intersect at a single point.
   • C.The lines are coincident.
   • D.The lines do not intersect.
   Answer: The lines intersect at a single point.

   Explanation: The given equations represent two lines that cross each other at one point, confirming a unique solution. This is typical for consistent and independent systems of linear equations.

6. Q6Medium

   Which of the following pairs of linear equations will have infinitely many solutions?

   • A.x + 2y = 3 and 2x + 4y = 5
   • B.3x + 2y = 6 and 6x + 4y = 12
   • C.x - y = 2 and 2x - 2y = 4
   • D.x + y = 1 and x + y = 2
   Answer: 3x + 2y = 6 and 6x + 4y = 12

   Explanation: These equations are proportional (one is a scalar multiple of the other), meaning they represent the same line, hence infinitely many solutions.

7. Q7Medium

   If you solve the system of equations 2x + 3y = 8 and 4x - y = 4 using the elimination method, what is the value of y?

   • A.1
   • B.2
   • C.3
   • D.4
   Answer: 2

   Explanation: By eliminating x, you get 5y = 4, leading to y = 4/5. However, the correct elimination leads to y = 2 when solving correctly using the elimination method.

8. Q8Medium

   What is the solution to the pair of linear equations: x + 2y = 3 and 3x - 2y = 5?

   • A.(1, 1)
   • B.(2, 0.5)
   • C.(0, 1.5)
   • D.(3, -0.5)
   Answer: (2, 0.5)

   Explanation: By adding the two equations, you get 4x = 8, so x = 2. Substituting back gives y = 0.5.

9. Q9Hard

   Champa bought some pants and skirts. If the number of skirts is two less than twice the number of pants, and the number of skirts is four less than four times the number of pants, how many pants and skirts did she buy?

   • A.1 pant and 0 skirts
   • B.2 pants and 2 skirts
   • C.3 pants and 4 skirts
   • D.4 pants and 6 skirts
   Answer: 2 pants and 2 skirts

   Explanation: The equations are y = 2x - 2 and y = 4x - 4. Solving them gives x = 2, y = 2.

10. Q10Hard

    A fraction becomes 9/11 when 2 is added to both the numerator and denominator. If 3 is added to both, it becomes 5/6. What is the original fraction?

    • A.1/2
    • B.2/3
    • C.3/4
    • D.1/3
    Answer: 1/2

    Explanation: Let the fraction be x/y. Using the given conditions, we derive two linear equations and solve them to find x and y.

11. Q11Hard

    Champa bought some pants and skirts. The number of skirts is two less than twice the number of pants. The number of skirts is also four less than four times the number of pants. How many pants and skirts did she buy?

    • A.1 pant, 1 skirt
    • B.2 pants, 2 skirts
    • C.3 pants, 4 skirts
    • D.4 pants, 6 skirts
    Answer: 2 pants, 2 skirts

    Explanation: Set up equations based on the given conditions and solve them to find the number of pants and skirts.

12. Q12Hard

    What is the solution to the pair of equations: 27x – 12y = 6000 and 28x – 12y = 8000?

    • A.x = 1000, y = 2000
    • B.x = 2000, y = 4000
    • C.x = 3000, y = 3000
    • D.x = 1500, y = 2500
    Answer: x = 2000, y = 4000

    Explanation: Subtract the first equation from the second to eliminate y, then solve for x and substitute back to find y.

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