Consider the system of equations 4x+14y=18 and 6x+21y=33.
For the first equation 4x+14y=18, dividing by 2:
2x+7y=9
For the second equation 6x+21y=33, dividing by 3:
2x+7y=11
The simplified equations are:
2x+7y=9
2x+7y=11
The left sides are identical, but the right sides are different: 9=11.
The same expression 2x+7y cannot equal both 9 and 11 simultaneously. These represent parallel lines that never intersect, so no point (x,y) satisfies both equations.
Testing x=1,y=1 in the original equations:
4(1)+14(1)=18 (satisfies first equation)
6(1)+21(1)=27=33 (does not satisfy second equation)
Similarly, x=2,y=1 does not satisfy both equations.
For infinite solutions, both sides would need to be identical after simplification, which is not the case here.
Therefore, the system has no solution (Option 4).