To determine the number of solutions for the given equation -12g + 9 = 2q - 6-15q, we can simplify it by grouping the q and g terms together and simplifying the constants:
-12g -15q = -15
Now, we can see that this is a linear equation in two variables. When such an equation is in this form, it can either have one unique solution, no solution, or infinitely many solutions, depending on the values of the coefficients.
We can solve for one variable in terms of the other and see if any constraints come up. So, we will solve for q in terms of g:
-12g -15q = -15 => 5q = -12g - 15 => q = (-12/5)g - 3
Since we have one variable in terms of another, substituting the above value of q in the original equation we get:
-12g + 9 = 2q - 6-15q => -12g + 9 = 2[(-12/5)g - 3] - 6-15[(-12/5)g - 3] => -12g + 9 = (-24/5)g + 24 => (-60/5)g = -15 => g = 1/4
Now, we can substitute the value of g in either of the two equation and get the value of q:
q = (-12/5)(1/4) - 3
=> q = -39/20
Therefore, the given equation has exactly one unique solution, namely (g,q) = (1/4,-39/20). Thus the answer is "one solution".