Final answer:
The equation of the line is y = (2/3)x + 2. The slope of the line is 2/3. The equation of the line perpendicular to this line and going through the point (4, -1) is y = (-3/2)x + 5.
Step-by-step explanation:
To express the equation 2x - 3y + 6 = 0 in slope and y-intercept form (y = mx + b), we need to isolate the y variable. Rearranging the equation, we get y = (2/3)x + 2. So the equation of the line in slope and y-intercept form is y = (2/3)x + 2.
The slope of this line is 2/3. The slope represents the steepness of the line and tells us that for every increase of 1 in the x-coordinate, the y-coordinate increases by 2/3.
A line perpendicular to the given line will have a slope that is the negative reciprocal of the slope of the given line. So the slope of a line perpendicular to this line is -3/2.
To determine the equation of the line perpendicular to the given line and going through the point (4, -1), we can use the point-slope form of a linear equation. The equation will be y - (-1) = (-3/2)(x - 4), which simplifies to y + 1 = (-3/2)x + 6. Rearranging the equation, we get y = (-3/2)x + 5.