To find the equation of a line perpendicular to the line 3x + 2y = -1 and passing through the point (5, 3), we need to determine the slope of the given line first.
The given line is in the form Ax + By = C, where A = 3, B = 2, and C = -1. To find the slope of this line, we can rearrange the equation in slope-intercept form (y = mx + b), where m is the slope:
3x + 2y = -1
2y = -3x - 1
y = (-3/2)x - 1/2
The slope of the given line is -3/2. Since a line perpendicular to this line will have a negative reciprocal slope, we can find the perpendicular slope by taking the negative reciprocal of -3/2:
Perpendicular slope = -1 / (-3/2) = 2/3
Now we have the slope of the perpendicular line, and we can use the point-slope form of a line (y - y₁ = m(x - x₁)) to find its equation. Plugging in the values (5, 3) for (x₁, y₁) and 2/3 for m:
y - 3 = (2/3)(x - 5)
Expanding and simplifying:
3y - 9 = 2x - 10
2x - 3y = 1
Therefore, the equation of the line that is perpendicular to 3x + 2y = -1 and passes through the point (5, 3) is 2x - 3y = 1.
To find the equation of a line parallel to the given line and passing through the point (5, 3), we can use the same method. Since parallel lines have the same slope, the slope of the parallel line will also be -3/2.
Using the point-slope form with (5, 3) and -3/2:
y - 3 = (-3/2)(x - 5)
Expanding and simplifying:
2y - 6 = -3x + 15
3x + 2y = 21
Therefore, the equation of the line that is parallel to 3x + 2y = -1 and passes through the point (5, 3) is 3x + 2y = 21.