What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2?
To find the $x$-intercept of the line, I need to know its equation. I can use the fact that it is perpendicular to the given line and has a $y$-intercept of 2 to find its equation. Perpendicular lines have slopes that are negative reciprocals of each other, so I can find the slope of the given line by rearranging it into slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. To do that, I subtract $3x$ from both sides and divide everything by $-2$, and I get $y = \frac{3}{2}x - 3$. So the slope of the given line is $\frac{3}{2}$, which means the slope of the perpendicular line is $-\frac{2}{3}$. Since I also know the $y$-intercept of the perpendicular line is 2, I can write its equation as $y = -\frac{2}{3}x + 2$. Now I can find the $x$-intercept of this line by setting $y$ to zero and solving for $x$. I get $0 = -\frac{2}{3}x + 2$, which implies $\frac{2}{3}x = 2$, which implies $x = 3$. So the $x$-intercept of the perpendicular line is 3. 