The pair of equation x + 2y + 5 = 0 and - 3x - 6y + 1 = 0 has


Given, equations are x + 2y + 5 = 0 and - 3x - 6y + 1 = 0.

Comparing the equations with general equation of the form:


ax + by + c = 0;


a1x + b1y + c1 = x + 2y + 5


a2x + b2y + c2 = - 3x - 6y + 1


Here, a1 = 1, b1 = 2, c1 = 5


And a2 = - 3, b2 = - 6, c2 = 1


Taking the ratio of coefficients to compare -


a1 /a2 = - 1/3


b1 /b2 = - 1/3


c1 /c2 = 5/1;


here a1/a2 = b1/b2 c1/c2


This represents pair of parallel lines.


Hence, the pair of equations has no solution.


Alternative solution -


We know for a line y = mx +c;


m represents slope of line.


So, we can write this lines in that form first -


For line: x + 2y + 5 = 0


2y = - 5 - x



Hence slope of first line is - 1/2.


For line: - 3x - 6y + 1 = 0


6y = - 3x + 1



Hence slope of second line is - 1/2.


Slope for both the lines represents the lines are parallel and will never intersect and so will have no solution.

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