To find the slope of a line that is perpendicular to the line given by the equation [tex]\(2x - y = 7\)[/tex], we need to perform the following steps:
1. Transform the given line's equation to slope-intercept form.
The given equation is:
[tex]\[
2x - y = 7
\][/tex]
We want to convert this to the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Start by isolating [tex]\(y\)[/tex]:
[tex]\[
-y = -2x + 7
\][/tex]
Multiply through by [tex]\(-1\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[
y = 2x - 7
\][/tex]
2. Identify the slope of the given line.
From the transformed equation [tex]\(y = 2x - 7\)[/tex], the slope [tex]\(m_1\)[/tex] of the given line is 2.
3. Determine the slope of the perpendicular line.
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. Therefore, if the slope of one line is [tex]\(m_1\)[/tex], the slope [tex]\(m_2\)[/tex] of the perpendicular line is given by:
[tex]\[
m_1 \cdot m_2 = -1
\][/tex]
Substituting [tex]\(m_1 = 2\)[/tex]:
[tex]\[
2 \cdot m_2 = -1
\][/tex]
Solve for [tex]\(m_2\)[/tex]:
[tex]\[
m_2 = -\frac{1}{2}
\][/tex]
4. Check the provided options and identify the correct slope.
The provided options are:
[tex]\[
\text{A. } -\frac{3}{2}
\][/tex]
[tex]\[
\text{B. } -\frac{1}{2}
\][/tex]
[tex]\[
\text{C. } \frac{1}{2}
\][/tex]
[tex]\[
\text{D. } \frac{3}{2}
\][/tex]
Comparing the calculated slope [tex]\(m_2 = -\frac{1}{2}\)[/tex] with the options, we see that the correct answer is:
[tex]\[
\text{B. } -\frac{1}{2}
\][/tex]
The slope of the line that is perpendicular to the line whose equation is [tex]\(2x - y = 7\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex]. The correct option is B.
