Answer:
y = 4/9x +10/3
Step-by-step explanation:
You want the slope-intercept equation of the line through point (-3, 2) that is perpendicular to the line y = -9/4x +1.
Slope-intercept equation
The slope-intercept form of the equation for a line is ...
y = mx + b . . . . . . m is the slope, b is the y-intercept
In order to write the desired equation, we need to know the desired slope and the y-intercept that makes the line go through the given point.
Slope
The slope of the perpendicular line is the opposite reciprocal of the slope of the given line. The given line equation is in slope-intercept form, so the coefficient of x is the slope of it: -9/4.
The slope of the perpendicular line is the opposite reciprocal of this:
m = -1/(-9/4) = 4/9
Y-intercept
Solving the slope-intercept form equation for b, we find ...
b = y - mx
Using the values of x and y for the given point, and the slope we just found, we have ...
b = 2 -(4/9)(-3) = 2 +4/3 = 10/3
Desired equation
The slope-intercept equation for a line with slope 4/9 and y-intercept 2/3 is ...
y = 4/9x +10/3
__
Additional comment
It can also be useful to start from the point-slope equation:
y -k = m(x -h) . . . . . line with slope m through point (h, k)
Your line is ...
y -2 = 4/9(x +3) . . . . . . use m=opposite reciprocal of -9/4; (h, k) = (-3, 2)
y = 4/9x +(4/9)(3) +2 = 4/9x +10/3 . . . . . . add 2 and simplify
