Answer:
The system will always have exactly one solution regardless of what you pick for 'a' and 'b'.
It is impossible to get "no solutions" or "infinitely many solutions" for this system.
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Step-by-step explanation:
Solve the first equation for y
2x - 3y = a
-3y = -2x + a
y = (2/3)x - a/3
The equation is in the form y = mx+c with m = 2/3 as the slope and c = -a/3 as the y intercept.
We'll compare this slope with the slope of the other line. Note how the parameter 'a' does not affect the slope at all. The slope of this line is always 2/3.
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Solve the second equation for y
x - 4y = b
-4y = -x+b
y = (1/4)x - b/4
The slope here is 1/4. This slope is different from the previous slope (2/3). Two lines having different slopes means they are not parallel. Non-parallel lines always cross at exactly one point. Like the first equation, this equation has the parameter 'b' not being part of the slope. The slope is always 1/4 for this equation.
Since 'a' and 'b' have nothing to do with the slopes, this means that the slopes are always the same. There is no way to get parallel lines. So we will never have a case of "no solutions". Also, we'll never get "infinitely many solutions" either. Why not? because infinitely many solutions only happens if the slopes are the same and the y intercepts are the same as well (ie the two lines are the same, usually in different form though).
Note: this system is consistent and independent. "consistent" means we have at least one solution. "independent" means one equation isn't a scalar multiple of the other. Saying "consistent and independent" together implies exactly one solution.