Question
find the gradients of a line perpendicular to 2y -3x = -5
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GPT-4o mini
To find the gradient (slope) of a line perpendicular to the line given by the equation \(2y - 3x = -5\), we first need to rearrange the equation into the slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope.
Starting with the equation:
\[
2y - 3x = -5
\]
we can solve for \(y\):
1. Add \(3x\) to both sides:
\[
2y = 3x - 5
\]
2. Divide every term by \(2\):
\[
y = \frac{3}{2}x - \frac{5}{2}
\]
From this, we can see that the slope (\(m\)) of the original line is \(\frac{3}{2}\).
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the gradient of the perpendicular line is:
\[
-\frac{1}{m} = -\frac{1}{\frac{3}{2}} = -\frac{2}{3}
\]
Therefore, the gradient of a line perpendicular to the line represented by the equation \(2y - 3x = -5\) is \(-\frac{2}{3}\).
Starting with the equation:
\[
2y - 3x = -5
\]
we can solve for \(y\):
1. Add \(3x\) to both sides:
\[
2y = 3x - 5
\]
2. Divide every term by \(2\):
\[
y = \frac{3}{2}x - \frac{5}{2}
\]
From this, we can see that the slope (\(m\)) of the original line is \(\frac{3}{2}\).
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the gradient of the perpendicular line is:
\[
-\frac{1}{m} = -\frac{1}{\frac{3}{2}} = -\frac{2}{3}
\]
Therefore, the gradient of a line perpendicular to the line represented by the equation \(2y - 3x = -5\) is \(-\frac{2}{3}\).
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