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

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

Example 1 :

3 x + 2 y = 5 and 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.

Example 2 :

2 x - 3 y = 8 and 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.

Example 3 :

(3/2) x + (5/3) y = 7 and 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.

Example 4 :

(4/3) x + 2 y = 8 and 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.

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