Finding Equation of Perpendicular Line through Point
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:
\[ y = -\frac{10}{3}x + 1 \]
