There are two positive integers a and b, where a b.

Question

There are two positive integers  𝑎  and  𝑏 , where  𝑎 > 𝑏 . The sum of the squares of these integers is equal to 5 times their sum. The difference of the squares of these integers is equal to 9 times their difference. What is the value of  𝑎 − 𝑏 ?
 
(A) 3 
(B) 4 
(C) 5 
(D) 6 
(E) 7

PyjamaScientist · Accepted Answer

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Alexabisaad · Answer

we have:

1. a^2 + b^2 = 5 ( a + b )
2. a^2 - b^2 = 9 ( a - b )

If we expand 2. we will have ( a + b )( a - b ) = 9 ( a - b )
we can cancel out (a-b) from both sides, then we will have (a+b) = 9

Replace it in 1. we will have a^2 + b^2 = 5 (9) = 45

Also a>b, so the possible combinations are either a=5,b=4 / a=6,b=3 / a=7,b=2 / a=7,b=1

See each one that satisfies both equations a+b = 9 and a^2 + b^2 = 45

We can see that 6^2+3^2=45 and 6+3=9.

Archit3110 · Answer

given
a>b
a^2+b^2 = 5*( a+b)
a^2-b^2= 9* (a-b)
add both
2*a^2 = 14a-4b
a^2 = 7a-2b
plugin values we see at a = 6 and b = 3 expression is valid
a-b= 3 correct option a

samarth1222 · Answer

hi 
In this question how did you plug in the value? I didn't understand the last step can you please explain that again

anuja3a · Answer

Please explain how you plugged in the values

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SBarros · Answer

"The difference of the squares of these integers is equal to 9 times their difference."

How do we know that this sentence translates to a^2 - b^2 = 9(a-b)

Why can't this translate to b^2 - a^2 = 9(b-a)

Since b < a, both sides of the equation would negative. To me, this seems like enough reason to try and solve this way too. What did you see in the question stem that showed you a-b was the correct way to address this problem.

Thank you for helping.

Bunuel · Answer

a^2 - b^2 = 9(a - b) and b^2 - a^2 = 9(b - a) are identical. You can multiply any of them by -1 to get the other.

luisdicampo · Answer

Deconstructing the Question

We are given two positive integers   a   and   b   with   a>b  , and

a^2+b^2=5(a+b)

a^2-b^2=9(a-b)

We need to find   a-b  .

The key idea is to simplify the second equation first using difference of squares.

Step-by-step

Start with

a^2-b^2=9(a-b)

Factor the left side:

(a-b)(a+b)=9(a-b)

Since   a>b  , we have   a-b 
e 0  , so divide both sides by   a-b  :

a+b=9

Now use the first equation:

a^2+b^2=5(a+b)=5\cdot 9=45

Use

(a+b)^2=a^2+b^2+2ab

Substitute:

9^2=45+2ab

81=45+2ab

36=2ab

ab=18

So   a   and   b   have sum   9   and product   18  .

Thus they are the roots of

x^2-9x+18=0

Factor:

(x-6)(x-3)=0

So

a=6,\ b=3

Therefore,

a-b=3

Answer: A

There are two positive integers a and b, where a > b.

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