HOW TO TELL IF A SYSTEM IS CONSISTENT OR INCONSISTENT

How to Tell if a System is Consistent or Inconsistent :

Consistent system of equations have at least one solution. 
Inconsistent system of equations have 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/b_2, 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 Nature of Solution of System of Linear Equations Examples

Example 1 :

On comparing the ratios a₁/a₂, b₁/b₂ and  c₁/c₂, find out whether the following pair of linear equations are consistent or inconsistent.

(i) 3 x + 2 y = 5

     2 x - 3 y = 7

Solution :

3 x + 2 y – 5 = 0

2 x - 3 y - 7 = 0

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  3, b₁  =  2, c₁  =  -5

a₂  =  2, b₂  =  -3, c₂  =  -7

a_1/a₂  =  3/2  -------(1)

b₁/b₂  = 2/3  -------(2)

c₁/c₂  =  -5/-7 = 5/7  -------(3)

This exactly matches the condition a₁/a₂ ≠  b₁/b₂.

Hence the system of equations is consistent.

(ii) 2 x - 3 y = 8

     4 x - 6 y = 9

Solution :

     2 x - 3 y – 8 = 0

     4 x - 6 y -9 = 0

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  2, b₁  =  -3, c₁  =  -8

a₂  =  4, b₂  =  -6, c₂  =  -9

a_1/a₂  =  2/4  =  1/2  -------(1)

b₁/b₂  = (-3)/(-6)  =  1/2 -------(2)

c₁/c₂  =  -8/(-9)  =  8/9 -------(3)

This exactly matches the condition a₁/a₂ = b₁/b₂ ≠ c₁/c₂

From this we can decide that the two lines are parallel. It means these two lines will not intersect each other. So it is inconsistent.

(iii)  (3/2) x + (5/3) y = 7

       9 x - 10 y = 14

Solution :

   (3/2) x + (5/3) y – 7 = 0

    9 x - 10 y – 14 = 0

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  3/2, b₁  =  5/3, c₁  =  -7

a₂  =  9, b₂  =  -10, c₂  =  -14

a_1/a₂  =  (3/2) / 9  =  1/6  -------(1)

b₁/b₂  = (5/3)/(-10)  =  -1/6 -------(2)

c₁/c₂  =  -7/(-14)  =  1/2 -------(3)

This exactly matches the condition a₁/a₂ ≠ b₁/b₂

From this, we can decide the two lines are intersecting. So it is consistent.

(vi) (4/3) x + 2 y = 8

      2 x + 3 y = 12

Solution :

  (4/3) x + 2 y – 8 = 0

   2 x + 3 y – 12 = 0

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  4/3, b₁  =  2, c₁  =  -8

a₂  =  2, b₂  =  3, c₂  =  -12

a_1/a₂  =  (4/3) / 2  =  2/3  -------(1)

b₁/b₂  =  2/3  -------(2)

c₁/c₂  =  -8/(-12)  =  2/3 -------(3)

This exactly matches the condition a₁/a₂ = b₁/b₂ = c₁/c₂

From this we may decide the two lines are coincident. So it is consistent.

After having gone through the stuff given above, we hope that the students would have understood, how to tell if a system is consistent or inconsistent.

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