Question

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

Answer 1

The system of equations Qx+Ry=S and Y=Tx+S has infinitely many solutions when the two equations represent the same line. This means that the slopes and y-intercepts of both equations are equal.

Step-by-step explanation:

The system of equations Qx+Ry=S and Y=Tx+S has infinitely many solutions when the two equations represent the same line. This means that the slopes and y-intercepts of both equations are equal.

To determine when the system has infinitely many solutions, we can equate the coefficients on both sides of each equation. This gives us the following conditions:

1. Q = T
2. R = 1
3. S = S

As long as these conditions are met, the system will have infinitely many solutions.

Answer 2

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.