If y = ax + b and y = cx + d for all values of x, where a, b, c, and d are constants, then all the following must be true EXCEPT:
A. a = c
B. ac = -1
C. a² = c²
D. |a| = √(c²)
E. ac + 1 > 0

(B) Since y is equal to both ax + b and cx + d, we know that: ax + b = cx + d.

This can only be true for all values of x if a = c and b = d.

So the two equations represent lines that have the same slope and the same y-intercept. Therefore, they represent the same line. Let us look at the choices one by one, to determine which is NOT NECESSARILY true.

Choice (A): We established above that this must be true.

Choice (B): This CANNOT be true. Since a and c must be the same number, their product cannot be negative.

Choice (C): This means that |a| = |c|. Since a = c, this must be true.

Choice (D): √(c²) is equal to |c|. So the equation in the answer choice becomes |a| = |c|, which is the same as the equation in choice (C), so it must be true.

Choice (E): Since a and c must be the same number, ac must be positive or 0 (0 if a = c = 0). So ac + 1 must definitely be positive.

We see that all the choices except choice (B) must be true. The correct answer is choice (B).
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Even option C could be the right answer!

Please, read original question statement very carefully. In this case it asks us to find the choice which is not necessarily true or is not true at all. We see that all the choices except choice (B) MUST BE TRUE. On the other hand, choice (B) is never true, so this is the answer we're looking for.

Furthermore, if choice (B) was right for some values of a, b, c, d, x and y, but it was not true for at least one possible value then choice (B) would also be the answer.

Let me extend the explanation
".... ax + b = cx + d. This can only be true for all values of x if a = c and b = d."

Indeed, the key for answering this question is finding that a = c. But why is it so?

Let us analyze
ax + b = cx + d

It must be true for every x. So let us plug in x = 0:
a × 0 + b = c × 0 + d
b = d

Knowing that we can plug b = d in equation ax + b = cx + d:

ax + d = cx + d
ax = cx
ax – cx = 0
x(a – c) = 0

Since it must be true for every x then we plug in x = 1:
1 × (a – c) = 0
a – c = 0
a = c

Therefore a = c MUST BE TRUE in any case and choice (A) is not an answer.

If a = c then |a| = |c|. Therefore choice (C) is also not an answer.