HOW TO TELL IF A SYSTEM IS CONSISTENT OR INCONSISTENT

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 + 2y = 8 and 2x + 3y = 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.

Example 5 :

5y = 10x + 11

−5y = 5x − 21

The solution to the given system of equations is (x,y). What is the value of 30x?

Solution :

5y = 10x + 11  ------(1)

−5y = 5x − 21  ------(2)

Applying the value of 5y from (1), we get

-(10x + 11) = 5x - 21

-10x - 11 = 5x - 21

-10x - 5x = -21 + 11

-15x = -10

x = 10/15

x = 2/3

Solving for 30x = 30(2/3)

= 20

So, the value of 30x is 20.

Example 6 :

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

(a) inconsistent     (b) consistent

(c) dependent consistent       (d) none of these

Solution :

2x + 3y = 5 ------(1)

4x + 6y = 10 ------(2)

3y = -2x + 5

y = (-2/3)x + 5/3

6y = -4x + 10

y = (-4x/6) + (10/6)

y = (-2x/3) + (5/3)

Since the above lines are having same slope and same y-intercept, the system of equations are consistent and it has infinitely many solution.

Example 7 :

The pair of equations y = 0 and y = –7 has

(a) one solution        (b) two solutions

(c) infinitely many solutions        (d) no solution

Solution :

y = 0 and y = –7

These lines are parallel, they will never meet. So, it has no solution.

Example 8 :

The pair of equations x = 4 and y = 3 graphically represents lines which are

(a) parallel             (b) intersecting at (3, 4)

(c) coincident        (d) intersecting at (4, 3)

Solution :

x = 4 is a line which is perpendicular to the x-axis.

y = 3 is the line which is parallel to the x axis.

The lines will intersect at the point (4, 3). Option d is correct.

Example 9 :

A pair of linear equations which has a unique solution x = 2, y = –3 is

(a) x + y = –1; 2x – 3y = –5

(b) 2x + 5y = –11; 4x + 10y = –22

(c) 2x – y = 1 ; 3x + 2y = 0

(d) x – 4y – 14 = 0; 5x – y – 13 = 0

Solution :

Option (a) :

x + y = –1; 2x – 3y = –5

Applying x = 2 and y = -3, we get

Option a is not system of linear equation which has the solution (2, -3).

Option (b) :

2x + 5y = –11; 4x + 10y = –22

Applying x = 2 and y = -3, we get

Option b is correct.

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