Questions: Through (-4,4); perpendicular to the line y=-(1/2) x+1

Transcript text: Through $(-4,4)$; perpendicular to the line $y=-\frac{1}{2} x+1$

Solution

Solution Steps

To find the equation of a line that passes through the point $(-4, 4)$ and is perpendicular to the line $y = -\frac{1}{2} x + 1$, we need to follow these steps:

Determine the slope of the given line.
Find the slope of the perpendicular line (negative reciprocal of the given slope).
Use the point-slope form of the equation of a line to find the equation of the desired line.

Step 1: Determine the Slope of the Given Line

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

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope $ m_{\perp} $ of the perpendicular line is: \[ m_{\perp} = -\frac{1}{-\frac{1}{2}} = 2 \]

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

The point-slope form of a line is given by: \[ y - y_1 = m (x - x_1) \] Given the point $(-4, 4)$ and the slope $ m_{\perp} = 2 $, we substitute these values into the point-slope form: \[ y - 4 = 2 (x + 4) \]

Step 4: Simplify the Equation

Simplify the equation to get it into slope-intercept form $ y = mx + b $: \[ y - 4 = 2x + 8 \] \[ y = 2x + 12 \]