To determine which line is perpendicular to a line with a slope of \( -\frac{1}{3} \), we need to find the slope of the perpendicular line.
The key fact to remember is that the slopes of two perpendicular lines multiply to give \(-1\).
1. Let's denote the slope of the line perpendicular to the given line as \( m_{\text{perpendicular}} \). Since the given line's slope is \( -\frac{1}{3} \), we have:
[tex]\[ m \times m_{\text{perpendicular}} = -1 \][/tex]
2. Substitute the given slope \( m = -\frac{1}{3} \) into the equation:
[tex]\[ -\frac{1}{3} \times m_{\text{perpendicular}} = -1 \][/tex]
3. To isolate \( m_{\text{perpendicular}} \), we can divide both sides of the equation by \( -\frac{1}{3} \):
[tex]\[ m_{\text{perpendicular}} = \frac{-1}{-\frac{1}{3}} \][/tex]
4. Simplifying the division:
[tex]\[ m_{\text{perpendicular}} = \frac{-1}{-\frac{1}{3}} = \frac{-1 \times (-3)}{1} = 3 \][/tex]
Thus, the slope of the line that is perpendicular to a line with a slope of \( -\frac{1}{3} \) is \( 3 \).
Therefore, the line that is perpendicular to a line with a slope of [tex]\( -\frac{1}{3} \)[/tex] is the one having the slope [tex]\( 3 \)[/tex].
