What is the slope of a line that is perpendicular to the line whose equation is [tex]2x - y = 7[/tex]?

A. [tex]-\frac{3}{2}[/tex]
B. [tex]-\frac{1}{2}[/tex]
C. [tex]\frac{1}{2}[/tex]
D. [tex]\frac{3}{2}[/tex]



Answer :

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.