NATURE OF SOLUTIONS OF LINEAR EQUATIONS IN TWO VARIABLES

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.

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