Quick way to determine number of integer solutions of $10xy + x + 7y = 2298$?

Question

Find how many integer solutions for $x,y$ does the following equation have? $$10xy + x + 7y = 2298$$

Is there a quick way? I’ve tried with SFFT but the $10xy$ term seems to be a problem when factoring.

Edit: This equation was developed as a quick way to obtain factoring decomposition of numbers ending in $7$. The solutions of the equation $(12,18)$ have the following pattern:

$127*181=22987$

$7$ and $1$ digits are added at the end of each solution, same with the $7$ added at the end of $2298$

Answer 1

One way is to use the fraction first, then remove the denominator.

$10xy+x+7y=2298$

$x(10y+1)+7y=2298$ , then create $(10y+1)$ for $7y$

$x(10y+1)+\frac{7}{10}(10y+1) - \frac{7}{10}=2298$

$(x + \frac{7}{10})(10y+1) = 2298+\frac{7}{10}$

now, remove the denominator $10$, by multiply 10 both sides.

$10(x + \frac{7}{10})(10y+1) = 10(2298+\frac{7}{10})$

$(10x + 7)(10 y + 1) = 22987 = 127\cdot 181 = 1 \cdot 22987$

so it means

\begin{equation} \left\{ \begin{aligned} 10x+7&=127\\ 10y+1&=181\\ \end{aligned} \right. \end{equation}

or

\begin{equation} \left\{ \begin{aligned} 10x+7&=22987\\ 10y+1&=1\\ \end{aligned} \right. \end{equation}

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The answer is

$(x,y) = (12,18) , (2298,0)$