Write the equation of the line L satisfying the given geometric conditions.
L has y-intercept (0,2) and is perpendicular to the line with equation
2x-3y=6
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3x+2y=4?
To find the equation of a line perpendicular to a given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rewrite the given equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
2x - 3y = 6
To isolate y, subtract 2x from both sides:
-3y = -2x + 6
Divide all terms by -3:
y = (2/3)x - 2
From this equation, we can see that the slope of the given line is 2/3.
Now, to find the slope of the line perpendicular to this, we take the negative reciprocal of the slope. The negative reciprocal of 2/3 is -3/2.
So, the slope of the line perpendicular to the given line is -3/2.
We also know that this perpendicular line passes through the y-intercept (0,2).
Using the point-slope form (y - y1 = m(x - x1)), where (x1, y1) is a point on the line and m is the slope, we can write the equation of the line:
y - 2 = (-3/2)(x - 0)
Simplifying:
y - 2 = (-3/2)x
To convert this equation to slope-intercept form, isolate y:
y = (-3/2)x + 2
Therefore, the equation of the line (L) that is perpendicular to the line 2x - 3y = 6 and passes through the y-intercept (0,2) is:
y = (-3/2)x + 2.