What I have so far...
x=3a-(3y/4)
But then what? I've tried playing around with all sorts of algebra but I never get closer to understanding why there is only 1 value of a that would result in only 1 possible solution for the system.
I do my best to not just solve math problems but to understand them, so I'll try and vomit what I think I know into words as best as I can.
The equation x^2+y^2=4 is a circle with radius 2 centered around the origin, and the equation (x/3)+(y/4)=a means that there is some relationship, that the constant a would define, between the two numbers that would only apply to a single point on the circle, or a specific x and a specific y value.
If I solve equation 1. for y I get y=(-(4x)/3)+4a which is a line with a slope of -4/3 with a y-intercept of 4a. When you plug in the answer of this problem (whatever it is) for a, the line will only intersect the circle at one point, hence there being only 1 possible solution for the system. For a circle and a line to only intersect at only one point, the line must be tangential to the circle. Also the circle and the line will have the same slope of -4/3 at the point of intersection.
So to find where they intersect I simply have to find where the circle has a slope of -4/3. But there are two answers for this because circles have 2 points with each possible slope value on opposite sides (point (1,0) and point (-1,0) share a slope of 0). In this situation because the line has a negative slope of -4/3, the circle would touch the line only once if it intersected at the point in quadrant 1 with a slope of -4/3 and also at the point in quadrant 3 that has a slope of -4/3. This means there are 2 values for a where there would be only one solution for the system but the question asks for a value of a where there is only 1 solution for the system of equations.
In other words, not only am I at a dead end but I think I have proved that there is NO answer to this question. I know that I am wrong because the question is multiple choice, but I don't see where I am wrong.