Pair of Linear Equations — Previous Year Questions (Class 10 Mathematics)
Pair of linear equations in two variables form the foundation of algebra. Mastering graphical and algebraic methods to solve these equations is essential f
TL;DR: Pair of linear equations in two variables form the foundation of algebra. Mastering graphical and algebraic methods to solve these equations is essent…
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Pair of linear equations in two variables form the foundation of algebra. Mastering graphical and algebraic methods to solve these equations is essential f
Pair of Linear Equations — Previous Year Questions with Solutions
Q (2023, 3 marks): Solve the pair of linear equations: 2x + 3y = 11 and x - 2y = -12 using substitution method.
Answer: From equation 2: x = 2y - 12
Substitute in equation 1: 2(2y - 12) + 3y = 11
4y - 24 + 3y = 11
7y = 35
y = 5
Substitute y = 5 in x = 2y - 12:
x = 2(5) - 12 = 10 - 12 = -2
Therefore, x = -2 and y = 5
Q (2022, 2 marks): For what value of k will the pair of linear equations 3x + 4y + 2 = 0 and 9x + 12y + k = 0 represent coincident lines?
Answer: For coincident lines, the ratios must be equal:
a1/a2 = b1/b2 = c1/c2
3/9 = 4/12 = 2/k
1/3 = 1/3 = 2/k
From 1/3 = 2/k:
k = 6
Q (2023, 5 marks): Graphically solve: x + y = 5 and 2x - y = 4. Also, find the area of the triangle formed by these lines and the y-axis.
Answer: Solving algebraically:
From x + y = 5: y = 5 - x
Substitute in 2x - y = 4:
2x - (5 - x) = 4
3x = 9
x = 3, y = 2
Intersection point: (3, 2)
Line 1 (x + y = 5) meets y-axis at (0, 5)
Line 2 (2x - y = 4) meets y-axis at (0, -4)
Triangle vertices: (0, 5), (0, -4), (3, 2)
Base = distance between (0, 5) and (0, -4) = 9 units
Height = perpendicular distance from (3, 2) to y-axis = 3 units
Area = (1/2) × 9 × 3 = 13.5 square units
Q (2021, 2 marks): Determine whether the pair of linear equations 4x - 6y + 3 = 0 and 2x - 3y + 1 = 0 is consistent or inconsistent.
Answer: For the pair a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
a1/a2 = 4/2 = 2
b1/b2 = -6/-3 = 2
c1/c2 = 3/1 = 3
Since a1/a2 = b1/b2 but a1/a2 ≠ c1/c2, the lines are parallel.
Therefore, the pair is inconsistent (no solution).
Q (2022, 3 marks): The sum of two numbers is 45 and their difference is 13. Find the numbers using a pair of linear equations.
Answer: Let the two numbers be x and y.
Equation 1: x + y = 45
Equation 2: x - y = 13
Adding both equations: 2x = 58, so x = 29
Substituting in equation 1: 29 + y = 45, so y = 16
The two numbers are 29 and 16.
Q (2023, 2 marks): For the pair of equations 2x + 3y = k and 4x + 6y = 2k + 3, find the value of k for which the system has infinitely many solutions.
Answer: For infinitely many solutions:
a1/a2 = b1/b2 = c1/c2
2/4 = 3/6 = k/(2k+3)
1/2 = 1/2 = k/(2k+3)
From 1/2 = k/(2k+3):
2k + 3 = 2k
3 = 0
This is impossible, so no value of k makes the system have infinitely many solutions.
Frequently Asked Questions
What is the difference between consistent and inconsistent pair of linear equations?
A consistent pair has at least one solution (either unique or infinite). An inconsistent pair has no solution. Graphically, consistent lines intersect or coincide, while inconsistent lines are parallel.
When do two linear equations represent the same line?
When the coefficients and constant terms are proportional, i.e., a1/a2 = b1/b2 = c1/c2, the equations represent coincident lines with infinitely many solutions.
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