The equation 4x + 3 = a(x + b), where a and b are constants, has no so

Question

The equation 4x + 3 = a(x + b), where  a  and  b  are constants, has no solutions. Which of the following must be true?

I.  a=4

II.  b=3

III.  b≠\frac{3}{4}

A) I only
B) I and II
C) I and III
D) I, II, and III
E) None of the above

Source: original question

AndrewN · Answer

Hello, everyone. So far, the question is proving to be harder than I had anticipated for the community, but the sample size is small. The keywords that stand out to me here are  no solutions . For this to be true, we need to show that the two sides of the equation CANNOT be equal. How do we do that? Start by distributing that  a :

4x+3=ax+ab

Notice the mirroring on each side of the equation: 4 x  would need to be equivalent to  ax , and 3 would need to be equivalent to  ab .

4x=ax

4=a

We know that  a   must  equal 4. Hence, we can strike answer (E). Now, applying the same logic as before, we can work on the remainder of the right-hand side:

3=ab

Since we know the value of  a , we can substitute:

3=(4)b

\frac{3}{4}=b

But now you have to be careful and remember the question. The problem tells us that the equation has  no  solutions, not an infinite number of them, so it  must  be true that although  a  = 4,  b  CANNOT be equal to 3/4. Hence, statements I and III must be true, and   the answer is (C)  .

If you were wondering why  b  could equal 3, you could test that value independently of  a , and you could easily create a workable equation (for instance, when  a  = 1).

I hope you had fun with this one. Good luck with your studies.

- Andrew

ischiragkapoor · Answer

hr1212   bb

Can someone explain why no solution means a=4 but b not equal to 3/4

Bunuel · Answer

For  no solution , the x-terms must cancel but the constant terms must not match.

4x + 3 = ax + ab

So we need a = 4. Otherwise, we could solve for x.

Then the equation becomes:

4x + 3 = 4x + 4b

For no solution, we need 3 ≠ 4b, so b ≠ 3/4.

If b = 3/4, both sides would be identical and there would be infinitely many solutions.

egmat · Answer

Hi ischiragkapoor,

The confusing part is exactly where Andrew's post first  derives  b =  3/4  and then, one line later, says b  cannot  equal  3/4 . That looks like a contradiction, but the two lines are answering two different questions.

What matching coefficients actually gives you

When Andrew set 4x + 3 = ax + ab and forced both sides to mirror each other ( a = 4  and  b = 3/4 ), he was building the case where the two sides are  identical . Identical sides means  every  x works - that's  infinitely many  solutions, not zero. So b =  3/4  is the value that would give you infinite solutions, which is precisely what the question does  not  want.

Split it into two clean steps

Move everything to one side:

- 4x + 3 = ax + ab → (4 - a)x = ab - 3

Now read the coefficient of x:

-  If a ≠ 4 , then (4 - a) isn't zero, so you can divide and get one clean value of x. That's a solution - not allowed. So  a must be 4 . (Statement I.)
-  With a = 4 , the x-term vanishes and you're left with 0 = 4b - 3, a plain number statement.

That last line decides everything:

- If 4b - 3 =  0  (i.e.  b = 3/4 ): you get 0 = 0, true for all x -  infinite  solutions.
- If 4b - 3 ≠  0  (i.e.  b ≠ 3/4 ): you get something like 0 =  5 , false for every x -  no  solution.

Since we need  no  solution, b must avoid  3/4 . That's Statement III.

A quick feel for it

- 2x + 3 = 2x + 5 - subtract 2x: 3 =  5 , false -  no solution .
- 2x + 3 = 2x + 3 - subtract 2x:  3  =  3 , true -  infinite solutions .

Same skeleton: once the x's cancel, a  false  leftover means no solution, a  true  leftover means infinitely many. That's why a =  4  and  b ≠  3/4  - answer  C .

Answer: C

Kinshook · Answer

The equation 4x + 3 = a(x + b), where  a  and  b  are constants, has no solutions. Which of the following must be true?

4x + 3 = ax + ab

If a = 4; 4x + 3 = 4x + 4b; b = 3/4; Infinite solutions
If a = 4 & b ≠ 3/4; Then the equation has no solution

Since it is given that the equation has no solution, both a = 4 & b ≠ 3/4 MUST BE TRUE

I.  a=4 
MUST BE TRUE

II.  b=3 
MAY OR MAY NOT BE TRUE

III.  b≠\frac{3}{4} 
MUST BE TRUE

IMO C

The equation 4x + 3 = a(x + b), where a and b are constants, has no so

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The equation 4x + 3 = a(x + b), where a and b are constants, has no so

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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My Rewards