Questions: Homework 1. Find the equation of the line that includes (4,7) and is perpendicular to the line, y=2/5 x-1

Transcript text: Homework 1. Find the equation of the line that includes $(4,7)$ and is perpendicuar to the line, $y=\frac{2}{5} x-1$

Solution

Solution Steps

Step 1: Determine the Slope of the Given Line

The slope of the given line $ y = \frac{2}{5}x - 1 $ is $ m_{\text{given}} = \frac{2}{5} = 0.4 $.

Step 2: Calculate the Slope of the Perpendicular Line

The slope of the line that is perpendicular to the given line is the negative reciprocal of $ m_{\text{given}} $: \[ m_{\text{perpendicular}} = -\frac{1}{m_{\text{given}}} = -\frac{1}{\frac{2}{5}} = -\frac{5}{2} = -2.5 \]

Step 3: Use the Point-Slope Form to Find the Equation

Using the point-slope form $ y - y_1 = m(x - x_1) $ with the point $ (4, 7) $: \[ b = y_1 - m_{\text{perpendicular}} \cdot x_1 = 7 - (-2.5) \cdot 4 = 7 + 10 = 17 \] Thus, the equation of the line can be expressed as: \[ y = -2.5x + 17 \]