Class 10 Maths Chapter 3 Question Answers - Pair of Linear Equations in Two Variables

Q1. Find the value of k so that graphs of
2x - ky = 9
6x - 9y = 18 will be parallel.
Sol.For the graph of equations of a system of linear equations to be parallel.

∴
⇒

Q2. Find the value of k so that the point (3, k) lies on the line represented by x - 5y = 5.
Sol. Substituting (3, k) in x - 5y = 5, we have:
3 - 5k = 5
⇒ - 5k = 5 - 3 = 2
⇒

Q3. Determine the value of k for which
kx + 3y = k - 3
12x + ky = k represent coincident lines.
Sol. ∵ For coincident lines,

∴
⇒
= 36 ⇒ k = 6
Also
⇒ 6k = k² ⇒ k = 6

Q4. If the system of linear equations,
3x + 2y - 4 = 0
px - y - 3 = 0 represents intersecting lines, then find p.
Sol.  a₁ = 3, b₁ = 2, c₁ = - 4
a₂ = p, b₂ = - 1, c₂ = - 3
For intersecting lines [i.e., having a unique solution]:

⇒

Q5. If (a + b) x + (2a - b) y = 21 and 2x + 3y = 7 has infinitely many solutions, then what is ‘a’ and ‘b’?
Sol. Here A₁ = (a + b), B₁ = (2a - b),
C₁ = - 21
A₂ = 2, B₂ = 3, C₂ = - 7
For infinite number of roots,

⇒
∴


⇒ a = 15/3 = 5
a + b =6 ⇒ b = 6 − 5 = 1
Thus, a = 5 and b = - 1

Q6. Write the relation between the coefficients for which the pair ax + by = c, lx + my = x has a unique solution.
Sol. A₁ = a, B₁ = b, C₁ = c
A₂ = l, B₂ = m, C₂ = n
For a unique solution,

i.e.,

Q7. Check if the pair of linear equations 3x + 6 = 10y and 2x - 15y + 3 = 0 is consistent or not?
Sol. Here, a₁ = 3, b₁ = - 10, c₁ = 6
a₂ = 2, b₂ = - 15, c₂ = 3
For the given pair of linear equations to be consistent,

⇒
⇒  which is true.
∴The given system of linear equations is consistent.

Q8. For what value of k, 
2x + 2y + 2 = 0 
4x + ky + 8 = 0 will have unique solution.
Sol. Here, a₁ = 2, b₁ = 2, c₁ = 2
a₂ = 4, b₂ = k, c₂ = 8
For the given system of linear equations to have a unique solution.

∴

Q9. Find the value of k for which the following system represents parallel lines:
 = 
2 (k - 1) x + y = 1
Sol. We have
=
and 2 (k - 1) x + y = 1
for parallel lines,

⇒
⇒
⇒ 3= − 2 (k − 1)
⇒ 3= − 2k + 2 

⇒ 3 − 2= − 2k
⇒ 1= − 2k
⇒

Q10. What is the point of intersection of the line 3x + 7y = 14 and the y-axis
Sol. ∵ For the point of intersection of a line and the y-axis, we put x = 0.
∴ 3x + 7y  = 14
⇒ 3 (0) + 7y  = 14
⇒ 7y = 14  ⇒ y = 2
∴ The point is (0, 2).

Q11. For what value of ‘a’ does the following pair of linear equations have infinitely many solutions?
4x - 3y - (a - 2) = 0,  8x - 6y - a = 0
Sol. We have:
4x - 3y - (a - 2) = 0
8x - 6y - a = 0
For infinitely many solutions, we have:

⇒
⇒
⇒
⇒ 2a − 4= a
⇒ a = 4

Q12. Find the number of solutions of the following pair of linear equations:
x + 2y - 8 =  0
2x + 4y =  16
Sol.We have:
x + 2y - 8 = 0
2x + 4y - 16 = 0
Here,

∴ The given system of equations has an infinite number of solutions.

Q13. Find the value (s) of ‘k’ for which the system of linear equations has no solutions.
kx + 3y = k - 2
12x + ky = k 

Sol.The given pair of linear equations are:
kx + 3y = k - 2     ...(1)
12x + ky  = k   ...(2)
For no solution of (1) and (2), we must have

i.e.,
⇒
i.e., k₂ = 36 and 3k ≠ k₂ − 2k
i.e., k = ± 6 and 3 ≠ k − 2
or k = ± 6 and k ≠ 5 ⇒ k = ± 6

Q14. Write whether the following pair of linear equations is consistent or not:
x + y = 14
x - y =  4  
Sol. Here,  
Since, for a consistent pair of linear equations,  , which is true for the given system [∵ 1 ≠ (- 1)]
Thus, it is a consistent pair of linear equations.

Q15. Find the value of k so that the following system of equations has no solution:
3x - y - 5 = 0
6x - 2y - k =  0
Sol. We have:
Here,

For no solution,
or
⇒

Q16. For what value of ‘a’, the point (3, a) lies on the line represented by 2x - 3y = 5?
Sol. Since, (3, a) lies on the equation
2x - 3y = 5
∴ (3, a) must satisfy this equation.
⇒ 2 (3) - 3 (a) = 5
⇒6 - 3a = 5
⇒ - 3a = 5 - 6 = - 1
⇒  
Thus the required value of a is 1/3.