To find the slope of the line that is perpendicular to the given line with the equation [tex]\(-2y = 3x + 7\)[/tex], we need to follow these steps:
1. Rewrite the equation of the given line in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
The original equation is:
[tex]\[
-2y = 3x + 7
\][/tex]
We want to express this in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
First, isolate [tex]\( y \)[/tex] by dividing every term by [tex]\(-2\)[/tex]:
[tex]\[
y = -\frac{3}{2}x - \frac{7}{2}
\][/tex]
2. Identify the slope of the given line:
From the equation [tex]\( y = -\frac{3}{2}x - \frac{7}{2} \)[/tex], we can see that the slope [tex]\( m \)[/tex] of the original line is:
[tex]\[
m = -\frac{3}{2}
\][/tex]
3. Determine the slope of the perpendicular line:
The slopes of perpendicular lines are negative reciprocals of each other. The negative reciprocal of a number [tex]\( a \)[/tex] is [tex]\(-\frac{1}{a}\)[/tex].
Thus, the negative reciprocal of the original slope [tex]\(-\frac{3}{2}\)[/tex] is:
[tex]\[
-\left(\frac{1}{-\frac{3}{2}}\right) = \frac{2}{3}
\][/tex]
Therefore, the slope of the line that is perpendicular to the given line is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
