Answer
$A = -\frac{3}{7}$
Work Step by Step
The slopes of two perpendicular lines multiply and result in $-1$. Therefore, we need to determine the slopes of both lines with the information given. In the case of the first equation: $$Ax + y - 2 = 0$$ $$y = -Ax + 2$$ where the slope is equal to $-A$. In the case of the second equation, of which we only know it contains the points $(1, -3)$ and $(-2, 4)$, we need to use the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$: $$m = \frac{4 - -3}{-2 - 1}$$ $$m = \frac{7}{-3}$$ We can now satisfy the first condition written above, to find $A$: $$-1 = -\frac{7}{3}(-A)$$ $$\frac{3}{7} = -A$$ $$-\frac{3}{7} = A$$
