Write an equation for line L in point-slope form and slope-intercept form.

Verified step by step guidance

1. Identify the slope of the given blue line. The equation is given as \(y = \frac{x}{2} - 2\), so the slope \(m_1\) is \(\frac{1}{2}\).
2. Since line L (red line) is perpendicular to the blue line, find the slope of line L. The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, \(m_2 = -\frac{1}{m_1} = -2\).
3. Determine a point on line L from the graph. From the image, the red line passes through the point \((2, -2)\).
4. Write the equation of line L in point-slope form using the point \((2, -2)\) and slope \(-2\): \(y - y_1 = m(x - x_1)\), which becomes \(y - (-2) = -2(x - 2)\).
5. Convert the point-slope form to slope-intercept form by simplifying and solving for \(y\): \(y = -2x + b\), where \(b\) is the y-intercept found by substituting the point into the equation.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the vertical change to the horizontal change between two points. It is often represented as 'm' in the equation y = mx + b. Understanding slope is essential for writing equations of lines and analyzing their relationships.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if the product of their slopes is -1. This means the slope of one line is the negative reciprocal of the other. Recognizing this relationship helps in finding the slope of a line perpendicular to a given line, which is crucial for writing its equation.

Point-Slope and Slope-Intercept Forms of a Line

The point-slope form, y - y1 = m(x - x1), uses a known point and slope to write a line's equation. The slope-intercept form, y = mx + b, expresses the line using slope and y-intercept. Both forms are fundamental for representing lines and converting between different equation formats.