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Maths / Linear Equations in Two Variables / Relationship of Coefficients with Number of Solutions

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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.

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Relationship of Coefficients with Number of Solutions

₹17200 ₹12000 +
Taxes