Question - Finding Equation of Perpendicular Line through Point
Solution:
To find the equation of the line h, which is perpendicular to line g and passes through the point (3, -9), we need to start by identifying the slope of line g.The equation for line g is written in slope-intercept form as:\[ y = \frac{3}{10}x - 8 \]The slope (m) of line g is the coefficient of x, which is $$\frac{3}{10}$$.For a line to be perpendicular to another line, its slope must be the negative reciprocal of the original line's slope. The negative reciprocal of $$\frac{3}{10}$$ is $$-\frac{10}{3}$$ (because when you flip the fraction and change the sign, you get the negative reciprocal).Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line to find line h's equation. Point-slope form is:\[ y - y_1 = m(x - x_1) \]where $$m$$ is the slope and $$(x_1, y_1)$$ is a point on the line.Using the slope of line h, which is $$-\frac{10}{3}$$, and the given point (3, -9), we plug these into the point-slope form:\[ y - (-9) = -\frac{10}{3}(x - 3) \]Simplify the equation:\[ y + 9 = -\frac{10}{3}x + \frac{10}{3} \cdot 3 \]\[ y + 9 = -\frac{10}{3}x + 10 \]We want to get it into slope-intercept form (y = mx + b), so we'll isolate y:\[ y = -\frac{10}{3}x + 10 - 9 \]\[ y = -\frac{10}{3}x + 1 \]The equation of line h in slope-intercept form, with numbers as simplified proper fractions, improper fractions, or integers, is:\[ y = -\frac{10}{3}x + 1 \]
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