If y = ax + b and y = cx + d for any value 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.

Since it is true for any value of x, let’s plug in x = 0. It yields b = d. If we plug in x = 1. It yields a + b = c + d. We already know that b = d, so a = c

You may think of the formulas as the two expressions representing the same line. That results in the fact that these expressions are the same.

Let’s look at the choices one by one, to determine which is NOT NECESSARILY true.

Choice (A): We have 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 + 1 must be positive.

The correct answer is B. ---------- Sorry, I don't understand why answer A (a = c) must be true. My logic follows like this: ax + b = cx + d ax – cx = d – b x(a – c) = d – b x = (d – b)/(a – c)

Since a – c could not equal 0, then a could not equal to c. Therefore a = c must not be true and A must be correct answer. Thanks for the explanation.

That is true, if we talk about x = (d – b)/(a – c), but would it be true if we considered the previous step x(a – c) = d – b ? Here, a – c can be 0. If we want to divide by it, we need to consider the case a – c = 0 separately. If a – c = 0, then 0 = d – b and the equality, 0 × x = 0, holds true for any x. That's exactly what we need, because the equality must be true for any value of x, while x = (d – b)/(a – c) gives us a constant instead of a variable, which can be any number, (of course, if a - c would not be 0).

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