Question

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 4 x plus yequals3 and y equals 4 x plus 3.
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Part 1
When does the first system of equations have infinitely many​ solutions?
When TequalsQ and RequalsS
When TequalsminusQ and Requals1
When TequalsQ and Requalsminus1
When TequalsminusQ and RequalsS

Answer 1

Substitute y = Tx + S into Qx + Ry = S to get (Q + RT)x + RS = S. For this to hold for every x on the line y = Tx + S we need Q + RT = 0 and RS = S. That is equivalent to Q = −RT and (R = 1 or S = 0).

From the choices given, the one that matches (with no extra restriction on S) is:
When T = −Q and R = 1.

Apply to the example: −4x + y = 3 and y = 4x + 3. Here Q = −4, R = 1, T = 4 and S = 3. T = −Q (4 = −(−4)) and R = 1, so the two equations are the same line (−4x + y = 3). Hence the system has infinitely many solutions.