A system with no solutions means that there is no possible values of x and y that will fulfill the equations
x-y=0 can be rearranged as x=y+0 which is the same as x=y which is the same as y=x. This system has an infinite amount of solutions, given that x and y have the same value
2x + 4y = 9 can be rearranged as 2x= 9 -4y which is the same as x= 4.5 -2y. We can use this value of x in the second equation:
4x + 2y = 9
4(4.5 -2y) +2y =9
18 - 8y + 2y =9
18-6y=9
-6y=-9
y=1.5
We now substitute y into the equation to find x
4x +2y =9
4x + 2(1.5) =9
4x + 3 =9
4x = 6
x=1.5
This system does have a solution of 1.5, 1.5
-5y = -4
y = 4/5
5x -y =-1
5x -(4/5) = -1
5x = -1 + 4/5
5x = 1/5
x= 1/25
This system has a solution of 1/25, 4/5
3x+y=5
y= 5 -3x
-3x -y= 2
-3x -(5 -3x) = 2
-3x -5 +3x =2
-3x + 3x = 7
0=7
This system does not have a solution
