SOLUTION: The sum of two numbers is eleven. The sum of the squares of the two numbers is sixty-five. Find the two numbers.

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Question 367744: The sum of two numbers is eleven. The sum of the squares of the two numbers is sixty-five. Find the two numbers.
Found 3 solutions by Alan3354, aprilburchett, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
Answer by aprilburchett(2) About Me  (Show Source):
You can put this solution on YOUR website!
First you must list all sums that will equal eleven:
10+1=11
9+2=11
8+3=11
7+4=11
6+5=11
Then you need to know which set of these numbers squared will equal 65:
10^2+1^2=100+1=101
9^2+2^2=81+4=85
8^2+3^3=64+9=73
7^2+4^2=49+16=65
6^2+5^2=36+25=61
You want the one that equals 65, so your two numbers are 6 and 5. They both add up to equal 11 and the sum of their squares is 65.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
system%28x%2By=11%2Cx%5E2%2By%5E2=65%29

Solve the first one for y

x%2By=11
y=11-x

Substitute %2811-x%29 for y in the second:

x%5E2%2By%5E2=65
x%5E2%2B%2811-x%29%5E2=65

Simplify and get 0 on the right side

x%5E2%2B%2811-x%29%2811-x%29=65
x%5E2%2B%28121-22x%2Bx%5E2%29=65
x%5E2%2B121-22x%2Bx%5E2=65
2x%5E2-22x%2B56=0

Divide every term by 2

x%5E2-11x%2B28=0

Factor the left side:

%28x-7%29%28x-4%29=0

Us the zero-factor principle:

x - 7 = 0          x - 4 = 0
    x = 7              x = 4

Substitute in 

y = 11 - x

If x = 7, then y = 11 - 7 or 4

If x = 4, then y = 11 - 4 or 7

So the numbers are 4 and 7.

Checking 4 + 7 = 11 and 4² + 7² = 16 + 49 = 65.  So those answers 
are correct.

Edwin