Consistency and Inconsistency of Linear Equations in Two Variables

**Consistency and Inconsistency of Linear Equations
in Two Variables**

Consider
linear equations in two variables say

*a*_{1}* x *+*
b*_{1}* y *+*
c*_{1}* *=* *0* *...(1)

*a*_{2}* x *+*
b*_{2}* y *+*
c*_{2}* *=* *0* *...(2)

where
*a* _{1} , *a*_{2} , *b*_{1}* *,*
b*_{2}* *,* c*_{1}* *and* c*_{2}*
*are real numbers.

Then
the system has :

**Example 3.54**

Check
whether the following system of equation is consistent or inconsistent and say how
many solutions we can have if it is consistent.

(i)
2*x* – 4*y* = 7; *x *– 3*y *= –2

(ii)
4*x* + *y* = 3 ; 8*x* + 2*y* = 6

(iii)
4*x* +7 = 2 *y ; *2*x* + 9 = *y*

*Solution*

**Example 3.55**

Check
the value of *k* for which the given system of equations *kx *+*
*2*y *=*
*3;* *2*x *−*
*3*y *=*
*1* *has a unique solution.

Given
linear equations are

*kx *+*
*2* y *=*
*3 .....(1)
[*a*_{1}* x *+* b*_{1}* y *+* c*_{1 }= 0]

2*x*
−
3*y* = 1 .....(2) [*a x *+* b y *+* c = 0 *]

Here
*a*_{1} = *k*, *b*_{1} =
2, *a*_{2} = 2, *b*_{2} = −3
;

For
unique solution we take *a*_{1}/*a*_{2} ≠ *b*_{1}/*b*_{2}
; therefore *k/2* ≠ 2/-3 ; *k* ≠ 4/-3 , that is k ≠ − 4/3 .

Find
the value of *k*, for the following system of equation has infinitely many
solutions. 2*x* − 3*y* =
7; (*k* + 2)*x* − (2*k* +
1)*y* = 3(2*k* −1)

Given
two linear equations are

2x − 3y = 7 [*a*_{1}* x *+ *b*_{1}* y *+ c_{1} = 0 ]

(*k* +
2)*x* − (2*k* + 1)*y* = 3(2*k* − 1) [*a*_{2}*x *+ *b*_{2}* y *+*
c*_{2} = 0 ]

Here
*a*_{1} = 2, *b*_{1} = −3, *a*_{2} = (*k* + 2), *b*_{2} = −(2*k* + 1), c_{1} = 7, c_{2} = 3(2*k* −1)

For
infinite number of solution we consider

Find
the value of *k* for which the system of linear equations 8*x* +
5*y* = 9; *kx *+* *10* y *=* *15* *has no solution.

** Solution** Given
linear equations are

8*x*
+
5*y* = 9 [ *a*_{1}* x *+*
b*_{1}* y *+*
c*_{1}* *=* *0]

*kx *+*
*10* y *=*
*15 [*a*_{2}* x *+*
b*_{2}* y *+*
c*_{2}* *=* *0]

Here
*a*_{1} = 8, *b*_{1} =
5, *c*_{1} = 9, *a*_{2} =
*k* , *b*_{2} = 10, *c*_{2} =
15

For
no solution, we know that

*a*_{1}/a_{2}
= *b*_{1}/*b*_{2} ≠ c_{1}/c_{2} and so, 8/*k* = 5/10 ≠ 9/15

80 =
5*k*

*k*= 16

1. Find the value of *k* for the given system of linear equations
satisfying the condition below:

(i) 2*x* + *ky* = 1; 3*x* − 5*y* = 7 has a unique solution

(ii) *kx *+* *3*y *=* *3; 12*x *+* ky *=* *6* *has no solution

(iii) (*k* − 3)*x* + 3*y* = *k*; *kx* + *ky* = 12 has infinite number
of solution

Solution:

2. Find the value of *a* and *b* for which the given system
of linear equation has infinite number of solutions 3*x* − (*a* + 1)*y* = 2*b* −1, 5*x* + (1 − 2*a*)*y* = 3*b*

Solution:

For the given linear equations, find another linear equation satisfying
each of the given condition

Solution:

Tags : Example Solved Problems | Algebra | Maths , 9th Maths : UNIT 3 : Algebra

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9th Maths : UNIT 3 : Algebra : Consistency and Inconsistency of Linear Equations in Two Variables | Example Solved Problems | Algebra | Maths

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