Problem: In the simultaneous equations
2xβ3y=8
6yβ4x=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 32β, and hence there is no intersection point. Algebraically, if
a2βx+b1βy=c1β and a2βx+b2βy=c2β,
then, multiplying the first equation by b2β, the second by b1β, and subtracting, we have
(a1βb2ββa2βb1β)x=c1βb2ββc2βb1β;β΄x=a1βb2ββa2βb1βc1βb2ββc2βb1ββ.
Thus if a1βb2ββa2βb1β=0 and c1βξ =0,c2βξ =0,x is undefined and no solution exists.