Question

Under what circumstances does the system of equations Qx+Ry=S and Y=Tx+S have infinitely many solutions?

Answer 1

From these -Tx+y=S. If -T=Q/R, then y=-Qx/R+S, so Ry=-Qx+RS, Qx+Ry=RS=S.
If R is not equal to 1, or S is non-zero, the equations are inconsistent, so there would be no solutions.
If R=1 there are an infinite number of solutions given by Qx+y=S, or y=S-Qx or y=S+Tx.
If S=0, Qx+Ry=0 or y=-Qx/R or y=Tx.