If y = ax + b and y = cx + d for all values of x, where a, b, c, and d

Question

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. a^2 = c^2

D.  |a| = \sqrt{c^2}

E. ac + 1 > 0

Bunuel · Accepted Answer

800score Official Solution:

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.

GMATinsight · Answer

Given  : y = ax + b and y = cx + d

i.e. ax + b = cx + d

i.e. a = c

therefore, a^2 = c^2

therefore,  |a| = \sqrt{c^2}

and since both a and c have same value and same sign so their product must be positive
i.e. ac > 0
hence ac> -1

Answer: Option B

deepak268 · Answer

Bunuel 
 ax + b = cx + d
x(a-c)=d-b
x=(d-b)/ (a-c)

if a=c then x becomes undefined, thats why I chose A. 
whats wrong here?

Bunuel · Answer

If a=c, then we cannot write the highlighted part but x*0 = d-b would still be perfectly valid equation, which would mean that d-b=0 and x can take any value.

CyberStein · Answer

a=b; because both are coefficients of x. 
assuming a = 1
Eliminate option a, because it is true. If option a is eliminated, then you must also eliminate option c; AV and Root both suggest +/- relationship; therefore the absolute value of a = c; eliminate d. Option E is eliminate because it can be true.

Option B suggests either a or b is a negative, but not both. Since a = b, therefore, B, ac = -1 cannot be true.
Answer is B

Princ · Answer

OA: B
y = ax + b and y = cx + d for all values of x, means that both lines are coincident.
So option that will imply that they are not coincident, will be the answer.
Product of slope of two non vertical perpendicular line is -1.
i.e if m1.m2=-1; then line y=m1x+C1 and line y=m2x+C2 will be perpendicular.
Option B : ac =-1 means that line y = ax + b and y = cx + d are perpendicular.
So it must not be not true.

Souravi99 · Answer

if a=10, x=3 and b=3
y=ax+b............33=(10*3)+3

if c=11, x=3, and d=0
y=cx+d............33=11*3+0

then a is not qual to c and b is not qual to d. (option a, option b both are correct) @bunnel
could u pls guide

Adarsh_24 · Answer

Another reason for B.
product of the slope of 2 lines are -1 if they are perpendicular to each other. I dont think the lines can be equal if they are perpendicular.

If y = ax + b and y = cx + d for all values of x, where a, b, c, and d

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If y = ax + b and y = cx + d for all values of x, where a, b, c, and d

Prep Toolkit

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My Rewards