A line passes through the points $P$ and $Q.$ If $P = (-8,2)$ and $Q = (1,7),$ then write the equation of this line in the form $Ax + By = C,$ where $A$, $B$, and $C$ are integers with greatest common divisor $1,$ and $A$ is positive.
The slope of this line is
\[\frac{7 - 2}{1 - (-8)} = \frac{5}{9}.\]The line passes through $(1,7),$ so the equation of the line is
\[(y - 7) = \frac{5}{9} (x - 1).\]Multiplying both sides by 9, we get
\[9y - 63 = 5x - 5.\]Thus, the equation is $5x - 9y = 58,$ so $A = \boxed{5},$ $B = -9,$ and $C = 58.$
We can find the equation of the line passing through $P$ and $Q$ using the point-slope formula.
The point-slope formula states that the equation of a line passing through the point $(x_1, y_1)$ with slope $m$ is given by $y - y_1 = m(x - x_1)$.
Let's find the slope of the line passing through $P$ and $Q$. The slope $m$ is given by the difference in $y$-coordinates divided by the difference in $x$-coordinates:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2}{1 - (-8)} = \frac{5}{9}.$$
Now, we can choose any point on the line to substitute into the point-slope formula. Let's choose the point $P(-8, 2)$:
\begin{align*}
y - y_1 &= m(x - x_1) \\
y - 2 &= \frac{5}{9}(x - (-8)) \\
y - 2 &= \frac{5}{9}(x + 8).
\end{align*}
Next, we can multiply both sides of the equation by $9$ to get rid of the fraction:
$$9(y - 2) = 5(x + 8).$$
Expanding both sides, we get:
$$9y - 18 = 5x + 40.$$
Rearranging the equation, we have:
$$5x - 9y = -58.$$
Therefore, the equation of the line passing through points $P$ and $Q$ in the desired form $Ax + By = C$, where $A$, $B$, and $C$ are integers with greatest common divisor $1$, and $A$ is positive, is:
$$\boxed{5x - 9y = -58}.$$
To write the equation of a line in the form $Ax + By = C$, we need to find the values of $A$, $B$, and $C$.
First, we need to find the slope of the line. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the coordinates of $P$ and $Q$ into this formula, we have:
\[m = \frac{7 - 2}{1 - (-8)} = \frac{5}{9}\]
So the slope of the line is $\frac{5}{9}$.
Next, we need to find the $y$-intercept of the line, which is the value of $y$ when $x = 0$. We can use the point-slope form of a linear equation:
\[y - y_1 = m(x - x_1)\]
Using the coordinates of point $Q$, $(1,7)$, we have:
\[y - 7 = \frac{5}{9}(x - 1)\]
Simplifying this equation, we get:
\[9y - 63 = 5x - 5\]
Rearranging this equation to the form $Ax + By = C$, we have:
\[5x - 9y = 58\]
So the equation of the line passing through points $P$ and $Q$ in the required form is $5x - 9y = 58$.