Problem 47 - A Graphical Approach to College Algebra

Question

Find the equation of the line satisfying the given conditions, giving it in slope-intercept form if possible. Through \((-5,7),\) perpendicular to \(y=-2\)

Step-by-Step Solution

Answer

The equation of the line is \(x = -5\).

Step 1: Identify the slope of the given line

The equation is given as \(y = -2\), which is a horizontal line. The slope of a horizontal line is 0.

Step 2: Determine the slope of the perpendicular line

The properties of perpendicular lines state that the product of their slopes is -1. Since the slope of the given line is 0, the perpendicular line must have an undefined slope, which indicates a vertical line.

Step 3: Understand the characteristics of a vertical line

Vertical lines have an undefined slope and are expressed in the form \(x = k\), where \(k\) is a constant indicating the x-coordinate of all points on the line.

Step 4: Find the equation of the vertical line

Since the line is vertical and passes through the point \((-5, 7)\), its equation will be \(x = -5\). This equation represents a vertical line through x-coordinate -5.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to write the equation of a line. It helps us quickly identify the slope and the y-intercept of the line. This form is written as: \[ y = mx + b \] Where:

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

This format is particularly useful because it clearly illustrates how steep the line is and where it intersects the y-axis. If the slope \( m \) is positive, the line rises as you move from left to right; if it's negative, the line falls.
For example, the line equation \( y = 2x + 3 \) tells us the slope is 2, and the line crosses the y-axis at 3.
In the exercise at hand, they inquired about finding an equation perpendicular to a horizontal line, which doesn't use slope-intercept form but knowing this form helps understand the concept of parallel and perpendicular lines.

Horizontal and Vertical Lines

Horizontal and vertical lines are a special category of lines that are relatively easy to understand once you grasp their basic properties.

Horizontal Lines

In the coordinate plane, horizontal lines are straight lines that run parallel to the x-axis. The equation of a horizontal line is always in the form: \[ y = c \] Where \( c \) is the constant y-coordinate of every point on the line. These lines have a slope of 0, meaning there is no vertical change, only horizontal. A great example from our exercise is the line \( y = -2 \).

Vertical Lines

Vertical lines run parallel to the y-axis. Their equations are in the form:\[ x = k \] Where \( k \) is the x-coordinate constant. Vertical lines have undefined slopes because they rise vertically without any horizontal movement. In the exercise provided, the perpendicular line through the point \( (-5, 7) \) was found as \( x = -5 \), indicating a vertical line.

Equation of a Line

Understanding the equation of a line is crucial when dealing with linear relationships and geometry. There are several forms to express the equation of a line, each providing unique insights or simplifications:

• Slope-Intercept Form: Already discussed, \( y = mx + b \) is useful for identifying slope and y-intercept.
• Point-Slope Form: This form, \( y - y_1 = m(x - x_1) \), is derived from a known point on a line \((x_1, y_1)\) and the slope \( m \).
• Standard Form: The equation \( Ax + By = C \) (where \( A, B, \) and \( C \) are integer constants) is convenient for calculations involving both x and y-coordinates directly.

The exercise at hand asked for the equation of a line perpendicular to an already horizontal line. Horizontal lines, like \( y = -2 \), interact with vertical lines, like \( x = -5 \), perpendicularly, thanks to their respective slopes (0 and undefined). Knowing different forms makes it easier to visualize these lines when tackling geometry problems.