SOLUTION: Find two numbers such that the smaller subtracted from the larger is 9 and the difference of the square of the larger subtracted from square of the smaller is 9.

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Question 1118978: Find two numbers such that the smaller subtracted from the larger is 9 and the difference of the square of the larger subtracted from square of the smaller is 9.
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
**






Equivalent system should be
and may be easier to solve.

Elimination Method should give

** The above solution may be mistaken. Description states, "the difference of the square of the larger subtracted from square of the smaller is 9. "; which should mean, and .
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


Let a be the larger number and b the smaller. Then

(1)
(2)

Dividing (2) by (1) gives

(3)

Then (1) and (2) with a little algebra give the answer:

a = 5; b = -4

If you are good with mental arithmetic, you might realize that the only time the squares of two integers differ by 9 is with . So the absolute values of the two numbers are 5 and 4; then to get a difference of 9 between the two numbers, you get the answers 5 and -4.

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Tutor @ikleyn is right; I didn't read the question carefully enough.

Since it is the smaller subtracted from the larger that gives a result of 9, the numbers are 4 and -5; not 5 and -4.
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
Find two numbers such that the smaller subtracted from the larger is 9 and
the difference of the square of the larger subtracted from square of the smaller is 9.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


            The correct answer is  4  and  -5.

            Both @greenestamps and @josgarithmetic produced wrong solutions and wrong answers. 


You are gven that

a - b = 9         (1)     (where "a" is the larger number)
b^2 - a^2 = 9     (2)


From eq(2), you have  

(b-a)*(a+b) = 9.   (3)


In (3), replace (b-a) by -9, since it is given (see (1) ). You will get

(-9)*(a+b) = 9,   or

a + b = -1.


Thus you have this system of two linear equations

a - b =  9,   (1)
a + b = -1    (4)


Add them. You will get

2a = 9 + (-1) = 8  ====>  a = 8/2 = 4.


Then from eq(4),  b = -1 - 4 = -5.


Check.    b^2 - a^2 = (-5)^2 - 4^2 = 25 - 16 = 9.   ! Correct !


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