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/a = c1/c2, there are infinitely many solutions.

(iii)  a1/a2  =  a1/a ≠  c1/c2, there is no solution

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, b =  -4, c1  =  8

a2  =  7, b =  6, c2  =  -9

a1/a2  =  5/7  ------(1)

b1/b =  -4/6  ------(2)

c1/c =  -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, b =  3, c1  =  12

a2  =  18, b =  6, c2  =  24

a1/a =  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, b =  -3, c1  =  10

a2  =  2, b =  -1, c2  =  9

a1/a2  =  6/2  =  3

b1/b =  -3/-1 =  3

c1/c =  10/9

Here, a1/a =  b1/b2  ≠  c1/c2

Hence it has no solution.

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