Question

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Part 1
Think About the Process  Under what circumstances does the system of equations Qx plus Ry equals Upper S and yequalsTxplusS have infinitely many​ solutions?
Use your result to make a conclusion about the system of equations negative 5 x plus yequals3 and y equals 5 x plus 3.
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Part 1
When does the first system of equations have infinitely many​ solutions?
When TequalsQ and Requalsminus1
When TequalsminusQ and Requals1
When TequalsminusQ and RequalsS
When TequalsQ and RequalsS

Answer 1

Substitute y = Tx + S into Qx + Ry = S:
Qx + R(Tx + S) = S → (Q + RT)x + RS = S.
For this to hold for all x (infinitely many solutions) we need Q + RT = 0 and RS = S, i.e.
T = -Q/R and (R − 1)S = 0. In particular, if R = 1 then T = −Q (this works for any S).

From the choices given, the correct one is:
When T equals −Q and R equals 1.

Apply to the specific system −5x + y = 3 and y = 5x + 3:
Q = −5, R = 1, T = 5 = −Q, S = 3, so the two equations are identical (y = 5x + 3) and there are infinitely many solutions.