Find the equations of lines parallel to and perpendicular to the line 3x - 2y = -1 and passing through the point (3, -2)
Rearranging,2y=3x+1 so m=3/2 since the line is parallel to the equation,it gradient is 3/2.so it equation is:y-(-2)=3/2(x-3) 2y+4=3x-9 2y=3x-13 if it is perpendicular,it gradient is -1/3/2=-2/3 so the equation:y-(-2)=-2/3(x-3) 3y+6=-2x+6 3y=-2x
To find the equations of lines parallel and perpendicular to the line 3x - 2y = -1 and passing through the point (3, -2), we need to determine the slope of the given line and then use that information to find the equations of the parallel and perpendicular lines.
Step 1: Find the slope of the given line.
To find the slope of the line 3x - 2y = -1, we rearrange the equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
3x - 2y = -1
-2y = -3x - 1
y = (3/2)x + 1/2
From the equation, we can see that the slope of the given line is 3/2.
Step 2: Find the equation of the line parallel to the given line.
Parallel lines have the same slope. Therefore, the slope of the parallel line will also be 3/2.
Using the point-slope form of a line (y - y1 = m(x - x1)), we can substitute the slope and the given point (3, -2) into the equation to find the equation of the parallel line.
y - (-2) = (3/2)(x - 3)
y + 2 = (3/2)x - (9/2)
y = (3/2)x - (9/2) - 2
y = (3/2)x - 9/2 - 4/2
y = (3/2)x - 13/2
Therefore, the equation of the line parallel to 3x - 2y = -1 and passing through the point (3, -2) is y = (3/2)x - 13/2.
Step 3: Find the equation of the line perpendicular to the given line.
Perpendicular lines have negative reciprocal slopes. The slope of the given line is 3/2, so the slope of the perpendicular line will be -2/3 (negative reciprocal).
Using the point-slope form of a line, we can substitute the slope and the given point (3, -2) into the equation to find the equation of the perpendicular line.
y - (-2) = (-2/3)(x - 3)
y + 2 = (-2/3)x + 2
y = (-2/3)x + 2 - 2
y = (-2/3)x
Therefore, the equation of the line perpendicular to 3x - 2y = -1 and passing through the point (3, -2) is y = (-2/3)x.