Find the equation for the line that passes through the point (−1,2), and that is perpendicular to the line with the equation 4/3x−2y=4/3.
the slopes of perpendicular lines are negative reciprocals
in point-slope form ... y - 2 = -3/2 (x - -1)
slope-intercept ... y = -3/2 x + 1/2
To find the equation of a line that is perpendicular to the given line, we need to use the fact that the slopes of perpendicular lines are negative reciprocals of each other.
First, let's rearrange the equation of the given line into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
4/3x - 2y = 4/3
-2y = -4/3x + 4/3
y = 2/3x - 2/3
The slope of the given line is 2/3.
If two lines are perpendicular, their slopes are negative reciprocals of each other. Let's find the negative reciprocal of 2/3:
Negative reciprocal of 2/3 = -3/2
So, the slope of the perpendicular line will be -3/2.
Now, we need to find the equation of the line that passes through the point (-1, 2) with a slope of -3/2.
We can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Using (-1, 2) and m = -3/2, we have:
y - 2 = -3/2(x - (-1))
y - 2 = -3/2(x + 1)
y - 2 = -3/2x - 3/2
y = -3/2x - 3/2 + 2
y = -3/2x - 3/2 + 4/2
y = -3/2x + 1/2
Therefore, the equation of the line that passes through the point (-1, 2) and is perpendicular to the line 4/3x - 2y = 4/3 is y = -3/2x + 1/2.