the sum of two positive integers is 60, and their positive difference is 26. what is the positive difference between the squares of the two integers?
larger ---- x
smaller ---- y
x+y = 60
x-y = 26
add them
2x = 86
x=43
43+y = 60
y = 17
difference between their squares
= 43^2 - 17^2 = 1560
Let's assume the two positive integers are x and y.
According to the given information:
x + y = 60 ----(Equation 1)
x - y = 26 ----(Equation 2)
To find the positive difference between the squares of the two integers, we need to calculate |x^2 - y^2|.
First, let's simplify Equation 1 by solving it for x:
x = 60 - y
Now substitute the value of x in Equation 2:
(60 - y) - y = 26
Simplify the equation:
60 - 2y = 26
Now, solve for y:
-2y = 26 - 60
-2y = -34
y = (-34) / -2
y = 17
Now substitute the value of y into Equation 1 to find x:
x + 17 = 60
x = 60 - 17
x = 43
Thus, the two positive integers are 43 and 17.
To find the positive difference between the squares of the two integers, we calculate |43^2 - 17^2|:
|1849 - 289| = 1560
Therefore, the positive difference between the squares of the two integers is 1560.
To find the positive difference between the squares of two integers, we first need to find the values of the integers.
Let's assume the two integers are x and y.
Given that the sum of the two positive integers is 60, we can write the equation:
x + y = 60 (Equation 1)
We are also told that the positive difference between the two integers is 26, so we can write another equation:
|x - y| = 26 (Equation 2)
We have two equations with two unknowns(x and y), so we can solve these equations simultaneously to find the values of x and y.
Let's start by eliminating the absolute value in Equation 2:
If x ≥ y, then |x - y| = x - y
If x < y, then |x - y| = -x + y
Since we are looking for positive integers, we can eliminate the negative sign and write:
x - y = 26 (Equation 3)
Now we have a system of two equations (Equation 1 and Equation 3) that we can solve simultaneously.
From Equation 1, we can rearrange the equation to express x in terms of y:
x = 60 - y
Substituting this value of x in Equation 3, we have:
60 - y - y = 26
Simplifying the equation:
60 - 2y = 26
Now, let's solve for y:
60 - 26 = 2y
34 = 2y
Dividing both sides by 2:
y = 17
We have found the value of y as 17. Substituting this value back into Equation 1 to find x:
x + 17 = 60
x = 60 - 17
x = 43
So, the two integers are 43 and 17.
Finally, to find the positive difference between the squares of these integers, we calculate:
(43^2) - (17^2) = 1849 - 289 = 1560
Therefore, the positive difference between the squares of the two integers is 1560.