nes Find an equation for the line with the civen properties. Expless ycur enswer using cither the geveral form or the slope intercept form of the equation of a ine Perpencicular to the ine \( y=-\frac{1}{2} x+1 \); containing the port \( (-2,-4) \)

To find the equation of a line that is perpendicular to the given line \( y = -\frac{1}{2}x + 1 \), we need to determine the slope of the perpendicular line. The slope of the given line is \(-\frac{1}{2}\), so the slope of the perpendicular line will be the negative reciprocal, which is 2. Now, using the point-slope form of the line equation, we can substitute the slope and the point \( (-2, -4) \) into the equation: \[ y - y_1 = m(x - x_1) \] \[ y - (-4) = 2(x - (-2)) \] This simplifies to: \[ y + 4 = 2(x + 2) \] \[ y + 4 = 2x + 4 \] \[ y = 2x + 0 \] So, the final line equation in slope-intercept form is \( y = 2x \) or you can express it in general form as \( 2x - y = 0 \). If you're ever solving for the equation of a line, keep in mind that understanding slope behavior is key! Perpendicular lines are like a dance, where the angles keep spinning into new directions, but there's always that mathematically chic negative reciprocal waiting to shine! Also, don't forget to plug-and-chug numbers for practice! You can play around with different points and slopes to build your confidence. The more you work with lines, the easier it becomes to visualize their relationships on a graph—it's like plotting out a treasure map where ‘X’ marks the spot for the hidden gems of math knowledge!