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 = k^{2} ⇒ k = 6
Q4. If the system of linear equations,
3x + 2y  4 = 0
px  y  3 = 0 represents intersecting lines, then find p.
Sol. a_{1} = 3, b_{1 }= 2, c_{1 }=  4
a_{2} = p, b_{2} =  1, c_{2} =  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 A_{1} = (a + b), B_{1} = (2a  b),
C_{1} =  21
A_{2} = 2, B_{2} = 3, C_{2} =  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. A_{1} = a, B_{1} = b, C_{1} = c
A_{2} = l, B_{2} = m, C_{2} = 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, a_{1 }= 3, b_{1} =  10, c_{1} = 6
a_{2} = 2, b_{2} =  15, c_{2} = 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, a_{1} = 2, b_{1} = 2, c_{1} = 2
a_{2} = 4, b_{2} = k, c_{2} = 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 yaxis
Sol. ∵ For the point of intersection of a line and the yaxis, 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., k_{2} = 36 and 3k ≠ k_{2} − 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.
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1. What is a pair of linear equations in two variables? 
2. How can we solve a pair of linear equations in two variables? 
3. What is the importance of solving a pair of linear equations in two variables? 
4. Can a pair of linear equations in two variables have more than one solution? 
5. How can we determine the consistency of a pair of linear equations in two variables? 

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