The difference of two numbers is 7. The difference of their squares is 259. What are the two numbers?

Question

i know that x-y=7 and I know that x^2-y^2=259. I'm just not sure how to fully solve.

Answer 1

Hi Taylor. We know x - y = 7 and x² - y² = 259. We can solve the first equation for x or y and then substitute the result into the second equation.

 

Solve for x:

 

x - y = 7

 

Add y to both sides of the equation:

 

x = y + 7

 

Substitute for x:

 

x² - y² = 259

 

Substitute for x:

 

(y + 7)² - y² = 259

 

Expand first term:

 

y² + 14y + 49 - y² = 259

 

Combine terms:

 

14y + 49 = 259

 

Subtract 49 from both sides:

 

14y = 210

 

Divide both sides by 14:

 

y = 15

 

Substitute for y in first equation:

 

x - y = 7

 

x - 15 = 7

 

Add 15 to both sides:

 

x = 22

 

Solution:

 

The two numbers are, x  = 22, and y = 15.

 

Check:

 

Substitute for x and y in both equations:

 

x - y = 7

 

22 - 15 = 7

 

x² - y² = 259

 

484 - 215 = 259

 

259 = 259, values check.

 

Questions?