4. Which equation represents a line perpendicular to the line whose equation is \( 2 x+3 y=12 \) ? \( \begin{array}{ll}\text { (1) } 6 y=-4 x+12 & \text { (3) } 2 y=-3 x+6 \\ \text { (2) } 2 y=3 x+6 & \text { (4) } 3 y=-2 x+12\end{array} \)

To find a line that is perpendicular to the line represented by the equation \( 2x + 3y = 12 \), we first need to determine its slope. Rewriting it in slope-intercept form (y = mx + b), we have \( 3y = -2x + 12 \) which simplifies to \( y = -\frac{2}{3}x + 4 \). The slope of this line is \( -\frac{2}{3} \). For a line to be perpendicular, its slope must be the negative reciprocal, which would be \( \frac{3}{2} \). Looking at the answer choices, the equation \( 2y = 3x + 6 \) (Option 2) can be solved to confirm that it has a slope of \( \frac{3}{2} \), making it the correct choice! Time to practice the art of identifying parallel and perpendicular lines! Remember, the key thing you want to do is look for those slopes: flip that fraction and change the sign! Feel free to play around with various line equations to hone your skills. It’s a fun way to engage with geometry—like creating your very own grid art with lines!