Relationship of Coefficients with Number of Solutions


 
 
Concept Explanation
 

Relationship of Coefficients with Number of Solutions

RELATIONSHIP OF COEFFICIENTS WITH NUMBER OF SOLUTIONS:

System of a pair of linear equations in two variables :

   large a_{1}x+b_{1}y+c_{1}=0

   large a_{2}x+b_{2}y+c_{2}=0

The given system of a pair of linear equations in tw variables has either one solution, infinite solutions or no solution.

large (i);;If;frac{a_{1}}{a_{2}}neq frac{b_{1}}{b_{2}},  then the system has one (or unique) solution, and the system intersect each other at only one point.

large (i);;If;frac{a_{1}}{a_{2}}= frac{b_{1}}{b_{2}}=frac{c_{1}}{c_{2}}, then the given system has infinte solution and the system is called dependent. In this case, a pair od lines represented by the system coincides with each other. So the intersect each other at infinite number of points.

large (i);;If;frac{a_{1}}{a_{2}}= frac{b_{1}}{b_{2}}neq frac{c_{1}}{c_{2}}, then the given system has no solution and hence the system is called inconsistent. In this case, a pair of lines represented by the system are parallel to each other.

So they do not intersect each other at any point.

Example : Check whether the pair of equation

                     large x+3y=6;;;;;....(i)

 and              large 2x-3y=12;;;;;....(ii)

SOLUTION 

                large frac{a_{1}}{a_{2}}=frac{1}{2},frac{b_{1}}{b_{1}}=frac{3}{-3}=-1;;;therefore frac{a_{1}}{a_{2}}neq frac{b_{1}}{b_{2}}

Hence the given system of a pair of equations in two variable is consistent.

Sample Questions
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Question : 1

By comparing the linear equation 2y + 3x + 4 = -2v+11 with ax + by + c = 0, find the value of - b

Right Option : B
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Explanation
Question : 2

A system of liner equations is given as follows :

a_1x+b_1y+c_1=0

a_2x+b_2y+c_2=0

Condition for two lines to have infinitely many solutions is ______________________________

Right Option : A
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Explanation
Question : 3

By comparing the linear equation - 3y + 2x - 9 = 0 with ax + by + c = 0, find the value of -c.

Right Option : A
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Explanation
 
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