Equation of Perpendicular Line Passing Through a Point
To find the equation of the line h, which is perpendicular to line g, and passes through the point (3, −9), we first need to determine the slope of line g.
The equation for line g is given by:
\[ y = \frac{3}{10}x - \frac{8}{10} \]
The slope of line g is the coefficient of x, which is \(\frac{3}{10}\).
Two lines that are perpendicular to each other have slopes that are negative reciprocals of one another. So, the slope (m) of line h will be:
\[ m = -\frac{1}{\text{slope of g}} = -\frac{1}{\frac{3}{10}} = -\frac{10}{3} \]
Now we know the slope of line h, we can use the point-slope form of the equation of a line to find its equation. The point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Plugging in our values for the slope and the coordinates of the given point (3, −9), we get:
\[ y - (-9) = -\frac{10}{3}(x - 3) \]
\[ y + 9 = -\frac{10}{3}x + 10 \]
To write this in slope-intercept form, which is \( y = mx + b \), we solve for y:
\[ y = -\frac{10}{3}x + 10 - 9 \]
\[ y = -\frac{10}{3}x + 1 \]
Here \( b \), the y-intercept, is 1. The numbers in the equation are already simplified proper fractions, improper fractions, or integers.
So, the equation for line h in slope-intercept form is:
\[ y = -\frac{10}{3}x + 1 \]
