SOLUTION: One positive number is 8 more than another. The sum of their squares is 130. Find both numbers.

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Question 41589: One positive number is 8 more than another. The sum of their squares is 130. Find both numbers.
Found 2 solutions by Nate, AnlytcPhil:
Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
Equation:
y%5E2+%2B+x%5E2+=+130
y%5E2+=+130+-+x%5E2
y+=+sqrt%28130+-+x%5E2%29
Other Equation:
y+=+x+%2B+8
Plug:
y+=+x+%2B+8
sqrt%28130+-+x%5E2%29+=+x+%2B+8
130+-+x%5E2+=+x%5E2+%2B+16x+%2B+64
0+=+2x%5E2+%2B+16x+-+66
0+=+x%5E2+%2B+8x+-+33
0+=+x%5E2+%2B+11x+-+3x+-+33
0+=+%28x%5E2+%2B+11x%29+%2B+%28-3x+-+33%29
0+=+x%28x+%2B+11%29+-+3%28x+%2B+11%29
0+=+%28x+-+3%29%28x+%2B+11%29
x+=+3 and x+=+-11
Plug:
y%5E2+%2B+3%5E2+=+130
y%5E2+%2B+9+=+130
y%5E2+=+121
y+=+11 or y+=+-11
The numbers are three and eleven.

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
One positive number is 8 more than another.
The sum of their squares is 130.  Find both numbers.

Let one positive number = x
Let another positive number = y

>>...One positive number is 8 more than another...<<

So                    x = y + 8

>>...The sum of their squares is 130...<<

So                  x² + y² = 130      

We have the system of equations

x = y + 8
x² + y² = 130

Substitute (y + 8) from the first for x in the second

           (y + 8)² + y² = 130

     (y + 8)(y + 8) + y² = 130

  y² + 8y + 8y + 64 + y² = 130
 
            2y² + 16y + 64 = 130

            2y² + 16y - 66 = 0

           2(y² + 8y - 33) = 0

          2(y + 11)(y - 3) = 0   

Set each variable factor = 0

        y + 11 = 0 gives y = -11

         y - 3 = 0 gives y = 3

Discard the negative answer
 
Substitute 3 for y in 

             x = y + 8

             x = 3 + 8

             x = 11

So the numbers are 11 and 3

Edwin
AnlytcPhil@aol.com