Let x represent the smaller positive integer and let y represent the larger integer.
Given:
One positive integer is 3 less than a second positive integer, we have the equation:
x = y - 3
The sum of the squares of the two integers is 65, we have the equation:
x² + y² = 65
Therefore, we have the set of equations:
x = y - 3.....................equation 1
x² + y² = 65.............equation 2
Since y represents the larger integer, let's find the value of y by solving both equations simultaneously using substitution method.
Substitute (y - 3) for x in equation 2:
x² + y² = 65
(y - 3)² + y² = 65
(y - 3)(y - 3) + y² = 65
y² - 3y - 3y + 9 + y² = 65
Combine like terms:
y² + y² - 3y - 3y + 9 = 65
2y² - 6y + 9 = 65
Subtract 65 from both sides:
2y² - 6y + 9 - 65 = 65 - 65
2y² - 6y - 56 = 0
Factor the left side of the equation:
2(y - 7)(y + 4) = 0
We have the factors:
y - 7 = 0 y + 4 = 0
Add 7 to both sides: Subtract 4 from both sides:
y - 7 + 7 = 0 + 7 y + 4 - 4 = 0 - 4
y = 7 y = -4
Thus, we have the values of y:
y = 7, y = -4
Since the interge