### Instruction: The graph of the equation $x + 2y + 3 = 0$ is perpendicular to the graph of the equation $ax + 2y + 3 = 0$. What is the value of $a$?### Response: To find the value of $a$, I need to compare the slopes of the two lines.
I can rewrite both equations in slope-intercept form by solving for $y$.
For the first equation, I get $y = -\frac{1}{2}x - \frac{3}{2}$.
For the second equation, I get $y = -\frac{a}{2}x - \frac{3}{2}$.
The slope of the first line is $-\frac{1}{2}$.
The slope of the second line is $-\frac{a}{2}$.
For the lines to be perpendicular, their slopes must be negative reciprocals of each other.
That means I need to find $a$ such that $-\frac{1}{2} = \frac{2}{a}$.
I can cross-multiply and get $-a = 4$.