two numbers differ by 5 and the sum of their squares is 17. Find the numbers
Answers
Hi,
Here is your answer
Let the two positive whole numbers be x and y.
Given that these numbers differ by = 5
Therefore x - y = 5
x = 5 + y
Also given that the sum of their squares = 193
Therefore
x² + y² = 193
Substituting the value of x
(5 + y)² + y² = 193
25 + y² + 10y + y ² = 193
2y² + 10y - 168 = 0
y² + 5y - 84 = 0
y² + 12y - 7y - 84 = 0
y(y + 12) - 7(y + 12)
(y + 12) (y - 7) = 0
y = -12 or y = 7
Rejecting y = -12 as given they are positive whole numbers.
Therefore y = 7
x - 7 = 5
x = 12
The two numbers are 12 and 7.
Hope this helps you.
Step-by-step explanation:
The sum of two numbers is 5 and the difference of their squares is 5. How would one find the difference between the numbers?
This test how well we can use algebra to find the values of 2 unknown numbers.
Let x = the first number greater than the second number. Let y be the second number.
The equations are: First for the numbers sum. x + y = 5 and next for the difference of their squares. x^2 - y^2 = 5.
We then write y in terms of x based on the first equation. y = 5 - x. We then substitute y in the second equation. x^2 - (5 - x)^2 =5
We then expand the binomial inside the parenthesis,
x^2 - (25 - 10x + x^2) = 5 and then remove the parenthesis
x^2 - 25 + 10x - x^2 = 5 and simplify terms
-25 + 10x = 5 then add 25 to both sides of the equation
10x = 30 and multiply both sides by 1/10
x = 3 and returning to x + y = 5 the value of y = 2 so the difference between the two numbers is 3 - 2 which simplifies to 1