If ax+by=17, 2ax+3by=43, and 3x=2y,

Hello Vincen.

I will try to explain to you this question in an easy way.

$$If\ \ \ ax+by=17\ \rightarrow\ -2ax-2by=-34.$$ Adding this equation to the second given equation we will get $$by=9.$$ Now, replacing this on the first given equation we get $$ax+by=17\ \leftrightarrow\ ax+9=17\ \leftrightarrow\ ax=8.$$ By the other hand, $$if\ \ \ \ 3x=2y\ \rightarrow\ \frac{x}{y}=\frac{2}{3}\ \leftrightarrow\ \frac{8}{9}=\frac{ax}{by}=\frac{2a}{3b}\leftrightarrow\ \frac{a}{b}=\frac{4}{3}.$$ As "a" and "b" are integers, we have the following cases:

$$\left(1\right)\ a=3\ and\ b=4.\ Then\ a+b=7.$$ $$\left(2\right)\ a=8\ and\ b=6.\ Then\ a+b=14.$$ $$\left(3\right)\ a=12\ and\ b=9.\ Then\ a+b=21.$$ $$\left(4\right)\ a=16\ and\ b=12.\ Then\ a+b=28.$$ $$\quad\hspace{2cm}\vdots$$ $$\left(k\right)\cdots\ a+b=7\cdot k.$$ If we see the options, we find that the correct answer is C .

I hope this explanation may help you.

Feel free to ask me if you have any doubt.

Cheers.