Question

Solve in positive integers, $14x-11y=29$

• A. $(5,0)$
• B. $(5,6)$
• C. $(6,5)$
• D. $(2,4)$

Answer 1

The correct option is C $(6,5)$

$14x-11y=29$ .........(i)

$\Rightarrow \frac{14x}{11}-y=\frac{29}{11}$

$\Rightarrow x+\frac{3x}{11}-y=2+\frac{7}{11}$

$\Rightarrow x-y+\frac{3x-7}{11}=2$

As $x$ and $y$ are positive integers

$\Rightarrow \frac{3x-7}{11}=$ integer

Multiplying by $4$, we get

$\Rightarrow \frac{12x-28}{11}=$ integer

$x-2+\frac{x-6}{11}=$ integer

$\Rightarrow \frac{x-6}{11}=$ integer

Let the integer be $p$

$\frac{x-6}{11}=p\Rightarrow x=11p+6$ .........(ii)

Substituting $x$ in (i), we get

$14(11p+6)-11y=29\Rightarrow 11y=154p+55\Rightarrow y=14p+5$ ........(iii)

From (ii) and (iii) we can see that the value of $x$ and $y$ are negative for integer $p<0$ , which is not possible as we are only dealing with positive integers.

So the values of $p$ can be $0,1,2,3,\mathrm{4.........}\infty$

Substituting $p$ in (ii) and (iii), we get the complete solution as

$\{\begin{array}{c}x=6,17,28,39,\mathrm{50..........}\infty \\ y=5,19,33,47,\mathrm{61..........}\infty \end{array}$

So, the equations have infinite solutions.