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Math Expert V
Joined: 02 Sep 2009
Posts: 58335
If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (01:43) correct 41% (02:02) wrong based on 264 sessions

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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

Kudos for a correct solution.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58335
If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

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.
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Re: If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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1
Bunuel wrote:
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. $$|a| = [square_rootc^2[/square_root]$$

E. ac + 1 > 0

Kudos for a correct solution.

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

i.e. ax + b = cx + d

i.e. a = c [Option A ruled out]

therefore, a^2 = c^2 [Option C ruled out]

therefore, $$|a| = \sqrt{c^2}$$ [Option D ruled out]

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 [Option E ruled out]

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Re: If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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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?
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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deepak268 wrote:
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?

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.
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If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

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.
Senior Manager  V
Joined: 22 Feb 2018
Posts: 420
If y = ax + b and y = cx + d for all values of x, where a, b, c, and d  [#permalink]

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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.
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Good, good Let the kudos flow through you If y = ax + b and y = cx + d for all values of x, where a, b, c, and d   [#permalink] 03 May 2018, 01:17
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