First, let's figure out the slope of the given line by writing the given line in slope-intercept form.
[tex]\frac{10}{3}y - 5x = -2 \\ \\ \frac{10}{3}y = 5x - 2 \\ \\ y = \frac{3}{10}(5x - 2) \\ \\ y = \frac{3}{2}x - \frac{3}{5}[/tex]
For a line in slope-intercept form, the coefficient of x is the slope of the line. The slope of the given line is 3/2.
You want a line perpendicular to the given line. Slopes of perpendicular lines are opposite recipricals. That means the numerator and denominator are flipped and the sign is changed.
The slope of a line perpendicular to the given line is -2/3.
You have the slope of -2/3 and also the given point (2, -9). To write the equation of the perpendicular line in slope-intercept form, you need the value of b. You can substitute these values into the formula for slope intercept form to find b, the y-intercept.
y = mx + b
-9 = (-2/3)(2) + b
b = -23/3
The equation of the line that goes through (2, 9) and is perpendicular to (10/3)y - 5x = -2 is y = (-2/3)x - 23/3
