SOLUTION: The sum of 2 numbers is 25. The sum of their squares is 313. Find the two numbers...thank you!

The sum of 2 numbers is 25. The sum of their squares is 313. Find the two numbers...thank you>...The sum of 2 numbers is 25...<<

x + y = 25

>>...The sum of their squares is 313...<<

x² + y² = 313

So you have the system of equations:

x + y = 25
x² + y² = 313
 
To solve by substitution:

Solve the first equation for y

y = 25 - x

Substityte (25 - x) for y in 

x² + y² = 313

x² + (25 - x)² = 313

x² + (625 - 50x + x²) = 313

x² + 625 - 50x + x² = 313

2x² - 50x + 625 = 313

2x² - 50x + 625 = 313

2x² - 50x + 312 = 0

2(x² - 25x + 156) = 0

Factor: two integers which multiply to give 156 and add to give 25
are 13 and 12.

2(x - 13)(x - 12) = 0

Set each factor = 0 and solve:

x - 13 = 0 gives x = 13

x - 12 = 0 gives x = 12

Then substitute each of these in 

y = 25 - x

Substituting x = 13

y = 25 - 13
 
y = 12

So one solution is (x, y) = (13, 12)

Substituting x = 12

y = 25 - 12
 
y = 13

So one solution is (x, y) = (12, 13)

Either way the two numbers are 12 and 13.

Edwin