Question

The number of triplets $(a,b,c)$ of the positive integer which satisfy the simultaneous equation $ab+bc=44$ & $ac+bc=23$ are:

Answer 1

The given equations are $ab+bc=44$ ..........(i)

And, $ac+bc=23$ .........(ii)

$\Rightarrow c(a+b)=23$, which is a prime number

So, the two factors must be $1$ & $23$.

$c=1$ and $a + b = 23$

$\Rightarrow b=23-a$
Put these values in (i), $b(a+c)=44$
$\Rightarrow (23-a)(a+1)=44$

$\Rightarrow 23a-a^2+23-a=44$
$\Rightarrow a^2-22a+21=0$

$\Rightarrow a^2-21a-a+21=0$
$\Rightarrow a(a-21)-1(a-21)=0$

$\Rightarrow (a-21)(a-1)=0$
$\Rightarrow a=1,21$

For, $a=1,b=22$

For $a=21,b=2$

$\therefore b=22,2$
The solution sets are $(1,22,1);(21,2,1)$.

Hence, the number of solutions sets $=2$.