Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

The equation of a line is given. Find the slope of a line that is a. parallel to the line with the given equation; and b. perpendicular to the line with the given equation. $$y=-\frac{2}{5} x-1$$

Step-by-Step Solution

Answer

The slope of the line parallel to the given line is \(-\frac{2}{5}\), and the slope of the line perpendicular to the given line is \(\frac{5}{2}\).

Step 1: Identify the slope of the given line

From the given equation \(y=-\frac{2}{5}x-1\), we can see that the slope \(m\) of the line is \(-\frac{2}{5}\).

Step 2: Find the slope of the line parallel to the given line

The slope of a line parallel to the given line is the same as the slope of the given line. Therefore, the slope of the line parallel to the given line is also \(-\frac{2}{5}\).

Step 3: Find the slope of the line perpendicular to the given line

The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. Therefore, the slope of the line perpendicular to given line is \(-\frac{1}{m}\), which equals \(\frac{5}{2}\).

Key Concepts

Parallel Lines and Their Slope

Parallel lines are two or more lines in a plane that never intersect. A key feature of parallel lines is that they have the same slope. This means if you know the slope of one line and you want to find the slope of a line parallel to it, it will be exactly the same. For example, if you have a line with the equation \(y = -\frac{2}{5}x - 1\), its slope \(m\) is \(-\frac{2}{5}\).

• To find a parallel line's slope, simply keep it the same: \( -\frac{2}{5}\).

This consistency in slope means that no matter how far they are extended, parallel lines will never cross each other. Understanding this concept simplifies many algebra problems, especially when dealing with equations of a line.

Perpendicular Lines and Their Slope

Perpendicular lines intersect at a right angle, which means they have a special relationship in their slopes. The slopes of two perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is \(m\), then the slope of a line perpendicular to it will be \(-\frac{1}{m}\). For instance, if one line has a slope of \(-\frac{2}{5}\), the perpendicular line's slope would be \(\frac{5}{2}\).

• To find the negative reciprocal, flip the fraction and change its sign.

This unique property allows for precise determination of perpendicular lines just by using their slopes, making it very simple to design right angles.

Understanding the Equation of a Line

The equation of a line in the slope-intercept form is \(y = mx + c\), where:

• \(m\) is the slope of the line.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

This form is highly useful for easily identifying the slope and intercept just by looking at the equation. If you have the equation \(y = -\frac{2}{5}x - 1\), then you know immediately:

• The slope \(m\) is \(-\frac{2}{5}\).
• The y-intercept \(c\) is \(-1\).

Grasping how to read a line's equation allows for swift comparison of lines, calculating intersection points, and determining relationships such as parallelism and perpendicularity.