Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. What is the slope of a line that is perpendicular to the line whose equation is \(A x+B y=C, A \neq 0\) and \(B \neq 0 ?\)

Step-by-Step Solution

Answer

The slope of a line that is perpendicular to the line whose equation is \(Ax + By = C\), where \(A \neq 0\) and \(B \neq 0\), is \(\frac{B}{A}\).

Step 1: Determine the slope of given line

From the equation of the given line, \(Ax + By = C\), where \(A \neq 0\) and \(B \neq 0\), we can convert this into slope-intercept form, \(y = mx + c\), to determine the slope. Do this by isolating \(y\) in terms of \(x\). In doing so, we get \(y = -\frac{A}{B}x + \frac{C}{B}\). Hence, the slope of the given line, \(m\), is \(-\frac{A}{B}\).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. This means if the slope of the given line is \(-\frac{A}{B}\), the slope of the line perpendicular to it, say \(m'\), is the negative reciprocal of \(-\frac{A}{B}\). Therefore, \(m' = -1/(-\frac{A}{B}) = \frac{B}{A}\). Thus, the slope of the line perpendicular to the given line is \(\frac{B}{A}\).

Step 3: Write the equation of the perpendicular line

Once we know the slope of the line that is perpendicular to the given line, we can write its equation in a slope-intercept form. Since the specific y-intercept isn't provided, let's denote it as \(c'\). The equation of the line is then \(y = m'x + c' = \frac{B}{A}x + c'\).

Key Concepts

Perpendicular Lines

Perpendicular lines are a key concept in geometry and algebra that help us understand the relationship between two lines in a plane. Two lines are said to be perpendicular if they intersect at a right angle, which is 90 degrees. This unique relationship between perpendicular lines has a specific impact on their slopes.

The slope of one line determines the slope of its perpendicular counterpart. If you know the slope of a given line, finding the slope of a line perpendicular to it is straightforward. Here’s the important part:

• The slope of the perpendicular line is the negative reciprocal of the original line's slope.

For instance, if a line has a slope of \(-\frac{A}{B}\), then the slope of any line perpendicular to it would be \(\frac{B}{A}\). This relationship allows for easy calculation and is fundamental when solving many geometric problems, including those involving right triangles and orthogonality.

Linear Equations

Linear equations represent straight lines and are foundational in algebra. They describe a one-to-one relationship between two variables, typically denoted as \(x\) and \(y\). A linear equation can be written in various forms, with the slope-intercept form \(y = mx + c\) being one of the most common and versatile.

In the slope-intercept form:

• \(m\) represents the slope of the line, indicating its steepness or incline.
• \(c\) is the y-intercept, showing where the line crosses the y-axis.

To determine the slope and y-intercept from a standard form equation, such as \(Ax + By = C\), you need to rearrange it into the slope-intercept form. This involves solving the equation for \(y\). For example, turning \(Ax + By = C\) into \(y = -\frac{A}{B}x + \frac{C}{B}\) reveals both the slope \(-\frac{A}{B}\) and the y-intercept \(\frac{C}{B}\). This form allows us to easily graph linear equations and compare different lines by their slopes and y-intercepts.

Slope of a Line

The slope of a line is a measure that describes both the direction and the steepness of the line. It plays a crucial role in the study of linear equations. Slope is typically denoted by \(m\) and calculated as the "rise over run," or the ratio of the vertical change to the horizontal change between two points on a line.

Mathematically, for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope \(m\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1} \]A positive slope indicates an upward trend as you move from left to right, while a negative slope points downward. If the slope is zero, the line is horizontal, and if undefined (division by zero), the line is vertical.

• A zero slope means the line is perfectly flat.
• An undefined slope means the line runs parallel to the y-axis.

Understanding the slope is fundamental not only in identifying the direction of a line but also in calculations involving geometric shapes and in determining the equations of lines that are either parallel or perpendicular to each other.