In Exercises 1–4, 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 line y = \frac{x}{2} - 2, which is \frac{1}{2}.
2. Since line L is perpendicular to the given line, calculate the negative reciprocal of \frac{1}{2}, which is -2.
3. Use the point-slope form of a line equation: y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
4. Substitute the slope -2 and a point from the graph, such as (0, -2), into the point-slope form: y - (-2) = -2(x - 0).
5. Convert the point-slope form to slope-intercept form by simplifying: y = -2x - 2.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing equations when you know a point on the line and its slope. It allows for easy identification of the slope and a specific point, making it straightforward to graph the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b is the y-intercept. This form is beneficial for quickly identifying the slope and where the line crosses the y-axis. It is commonly used for graphing linear equations and understanding the relationship between the variables.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m, the slope of the line perpendicular to it will be -1/m. Understanding this relationship is crucial for finding the equation of a line that is perpendicular to a given line, as it allows you to determine the necessary slope for the new line.