NATURE OF SOLUTIONS OF LINEAR EQUATION IN TWO VARIABLES
Nature of Solutions of Linear Equations in Two Variables :
In this section, we will learn how to find the nature of solutions of linear equations in two variables.
By solving any linear equation in two variables, we may have the following solutions.
(i) Unique solution
(ii) Infinitely many solution
(iii) No solution.
To apply the concept given below, the given equations will be in the form
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
(i) a1/a2 ≠ b1/b2, we get a unique solution
(ii) a1/a2 = a1/a2 = c1/c2, there are infinitely many solutions.
(iii) a1/a2 = a1/a2 ≠ c1/c2, there is no solution
Discussing the Nature of Solution of Linear Equations in Two Variables Examples
Example 1 :
On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
(i) 5 x – 4 y + 8 = 0
7 x + 6 y – 9 = 0
Solution :
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 5, b1 = -4, c1 = 8
a2 = 7, b2 = 6, c2 = -9
a1/a2 = 5/7 ------(1)
b1/b2 = -4/6 ------(2)
c1/c2 = -8/9 ------(3)
(1) ≠ (2)
Here, a1/a2 ≠ b1/b2
Hence it has unique solution.
(ii) 9 x + 3 y + 12 = 0
18 x + 6 y + 24 = 0
Solution :
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 9, b1 = 3, c1 = 12
a2 = 18, b2 = 6, c2 = 24
a1/a2 = 9/18 = 1/2 ------(1)
b1/b2 = 3/6 = 1/2 ------(2)
c1/c2 = 12/24 = 1/2 ------(3)
(1) = (2) = (3)
Here a1/a2 = b1/b2 = c1/c2
The given lines are having infinitely many solution.
(iii) 6 x - 3 y + 10 = 0
2 x - y + 9 = 0
Solution :
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 6, b1 = -3, c1 = 10
a2 = 2, b2 = -1, c2 = 9
a1/a2 = 6/2 = 3
b1/b2 = -3/-1 = 3
c1/c2 = 10/9
Here, a1/a2 = b1/b2 ≠ c1/c2
Hence it has no solution.
After having gone through the stuff given above, we hope that the students would have understood, nature of solutions of linear equations in two variables.
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