How do I solve this simultaneous equation for x and y: ax+by=c bx+ay=c Thank you for the help!

Kindly find the attached file for the answer.
 
Regards

Hello!
The most straightforward way is to use Cramer's Rule. The main determinant of this system is D = |[a,b],[b,a]| = a^2-b^2 = (a-b)(a+b). If it is nonzero, the system has the only solution. The determinant for the variable x is D_x =|[c,b],[c,a]| = c(a-b), the determinant for the variable y is also D_y =|[a,c],[b,c]| = c(a-b).
By the rule x = D_x/D = (c(a-b)) / ((a-b)(a+b)) = c/(a+b), y = D_y/D = c/(a+b).
 
[the remaining options are probably less interesting but we have to consider them]
If the main determinant D is zero, the system has many or no solutions.
If a = b, then both equations are the same, ax+ay=c. If a is nonzero, then the general solution is x=t, y=c/a-t. If a is zero and c is nonzero, there are no solutions. If a is zero and c is zero, then any pair of numbers is a solution.
If a = -b, we get equations ax-ay=c, -ax+ay=c, or a(x-y)=c=-c. If c is nonzero, then there are no solutions. If c is zero and a is nonzero, the general solution is x=t, y=t. If c is zero and a is zero, then any pair of numbers is a solution.