Express the equation of the line passing through points $(1,7)$ and $(2,-3)$ in the form $ax+by=c$, where $a$, $b$ and $c$ are integers, the greatest common divisor of $a$ and $b$ is $1,$ and $c

Question

Express the equation of the line passing through points $(1,7)$ and $(2,-3)$ in the form $ax+by=c$, where $a$, $b$ and $c$ are integers, the greatest common divisor of $a$ and $b$ is $1,$ and $c<0$.

GPT 3.5 · Answer

Using the two-point form, the equation of the line passing through $(1,7)$ and $(2,-3)$ is \[(y - 7) = \left(\frac{7 - (-3)}{1 - 2}\right)(x - 1).\]
Simplifying, we have \[y - 7 = 10(x - 1).\] Expanding the product on the right and rearranging the terms, we obtain \[10x - y = -3.\] Thus, the equation of the line in the desired form is $\boxed{10x - y = -3}$.