Problem: In the simultaneous equations
2x−3y=8
6y−4x=9:
Answer Choices:
A. x=4,y=0
B. x=0,y=3/2
C. x=0,y=0
D. there is no solution
E. there are an infinite number of solutions
Solution:
Geometrically, the problem represents a pair of parallel lines, with slope equal to 32​, and hence there is no intersection point. Algebraically, if
a2​x+b1​y=c1​ and a2​x+b2​y=c2​,
then, multiplying the first equation by b2​, the second by b1​, and subtracting, we have
(a1​b2​−a2​b1​)x=c1​b2​−c2​b1​;∴x=a1​b2​−a2​b1​c1​b2​−c2​b1​​.
Thus if a1​b2​−a2​b1​=0 and c1â€‹î€ =0,c2â€‹î€ =0,x is undefined and no solution exists.