To determine the slope of a line that is perpendicular to a line with a given slope, we'll use the concept of negative reciprocals.
1. The slope of the given line is \( \frac{1}{2} \).
2. For two lines to be perpendicular, the product of their slopes must be \(-1\).
3. Let \( m_1 = \frac{1}{2} \) be the slope of the first line. Let \( m_2 \) be the slope of the line we are looking for. The relationship between their slopes will be:
[tex]\[
m_1 \cdot m_2 = -1
\][/tex]
4. Substitute \( \frac{1}{2} \) for \( m_1 \):
[tex]\[
\left( \frac{1}{2} \right) \cdot m_2 = -1
\][/tex]
5. Solve for \( m_2 \) by multiplying both sides of the equation by \( 2 \):
[tex]\[
m_2 = -1 \times 2
\][/tex]
6. Simplifying the right side gives:
[tex]\[
m_2 = -2
\][/tex]
Therefore, the slope of a line that is perpendicular to a line with a slope of [tex]\( \frac{1}{2} \)[/tex] is [tex]\( -2 \)[/tex].
