How do you find two numbers such that their sum is #9# and the difference of their squares is also #9#?

Answer 1

4 and 5 (By trial and error)

You should try numbers such as 1 and 2 (their sum is 3 not 9), or 3 and 4 (their sum is 7 not 9). The other must is the difference of their squares. Therefore, you can reach your conclusion.

#4+5=9#

#5^2-4^2=25-16=9#

Your numbers are 4 and 5.

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Answer 2

Eva Abramson

#5# and #4#

Suppose that the two numbers are #a# and #b#. Then, from the question, it states that #{(a+b=9),(a^2-b^2=9):}#.

Divide the second equation by the first equation, or #(a^2-b^2)/(a+b)=9/9#, or #a-b=1#.

Thus, we have #{(a+b=9),(a-b=1):}#. Add these two equations to get #2a=10#, or #a=5#. Since #a+b=9#, #b=4#.

The answer is #5# and #4#.

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Answer 3

Avery Alexander

The numbers are #5 and 4#

Let the numbers be #x and y#

A: #x+y =9" "# their sum is #9#

B: #x^2 - y^2 =9" "# the difference of their squares is #9#

Solve for #x# in equation A

#color(blue)(x = (9-y))#

Substitute for #x# in equation B.

#" "color(blue)(x^2) - y^2 =9# #" "darr# #color(blue)((9-y))^2 - y^2 =9 #

#81-18y +y^2 -y^2 =9#

#81-9 = 18y#

#72 = 18y#

#y=4#

If #y=4" "rarr x = 5#

Check:

#5+4=9" and "25-16 = 9#

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Answer 4

Kennedy Adams

Let the two numbers be (x) and (y). We are given that their sum is 9, so (x + y = 9), and the difference of their squares is 9, so (x^2 - y^2 = 9).

From the first equation, we can express (y) in terms of (x): (y = 9 - x).

Substitute (y) in terms of (x) into the second equation:

(x^2 - (9 - x)^2 = 9)

Expand the squared term:

(x^2 - (81 - 18x + x^2) = 9)

Simplify:

(x^2 - 81 + 18x - x^2 = 9)

Combine like terms:

(18x - 81 = 9)

Add 81 to both sides:

(18x = 90)

Divide by 18:

(x = 5)

Now, substitute (x = 5) back into (y = 9 - x):

(y = 9 - 5)

(y = 4)

So, the two numbers are 5 and 4.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.