Question

The sum of the squares of two positive integers is 106 and the difference of the squares of the numbers is 56. Find the numbers.

Group of answer choices

Answer 1

To find the two positive integers, we can solve a system of equations using the given information. By assigning variables and manipulating the equations, we find that the numbers are 9 and 5.

Step-by-step explanation:

To solve this problem, let's assign variables to the two positive integers. Let's call them x and y. According to the given information:

x^2 + y^2 = 106

x^2 - y^2 = 56

Now, we can solve these equations simultaneously. Let's subtract the second equation from the first to eliminate the y^2 term:
x^2 + y^2 - (x^2 - y^2) = 106 - 56

Simplifying the left side:
2y^2 = 50

Dividing both sides by 2:
y^2 = 25

Taking the square root of both sides:
y = 5 or -5 (but we are looking for positive integers)

Now, substitute the value of y into one of the original equations to solve for x:
x^2 + 25 = 106

Simplifying:
x^2 = 81

Taking the square root of both sides:
x = 9 or -9 (again, we are looking for positive integers)

Therefore, the two positive integers are 9 and 5.