MHB Find $a,b,k$: Solving Equations

[Albert1]

given :

$a-b=k---(1)$

$a^2-b^2=70---(2)$

where $k\in N$ , and $0<b<1$

please find the values of $a,b ,k$

 

[kaliprasad]

given :

$a-b=k---(1)$

$a^2-b^2=70---(2)$

where $k\in N$ , and $0<b<1$

please find the values of $a,b ,k$

Spoiler

from the second equation as $0 < b < 1$ so $70<a^2<71$ hence the value of taking integers between 8 and 9
so we have
$8 < a < 9$ and k is integer part of a so k = 8
$a = 8 + b$
so we get
$a^2-b^2 = (8+b)^2-b^2 = 64 + 16b= 70$ or $b = \dfrac{3}{8}$
hence $k=8,b= \dfrac{3}{8}, a = 8\dfrac{3}{8}$