Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, 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 \(x = -5\), a vertical line through \((-5, 7)\).

Step 1: Identify the Equation of the Given Line

The given line is represented by the equation \(y = -2\). This indicates a horizontal line at \(y = -2\).

Step 2: Understand Perpendicular Line Characteristics

A line perpendicular to a horizontal line must be a vertical line. Vertical lines have undefined slopes and are expressed as \(x = a\), where \(a\) is a constant.

Step 3: Determine the Equation of the Perpendicular Line

Since the line passing through \((-5, 7)\) is perpendicular to \(y = -2\), it must be a vertical line. Thus, the equation of the line is \(x = -5\).

Step 4: Verify the Perpendicular Line Equation

Check that the equation \(x = -5\) passes through the point \((-5, 7)\), which it does, as any vertical line \(x = -5\) will contain all points where \(x = -5\). However, vertical lines cannot be expressed in slope-intercept form \(y = mx + b\), as their slope is undefined.

Key Concepts

Perpendicular Lines

In the realm of geometry, understanding perpendicular lines opens the door to uncovering numerous relationships within coordinate systems. When two lines are perpendicular, they intersect to form a right angle, precisely 90 degrees.

There are a few simple aspects to remember:

• Horizontal lines have zero slope.
• Vertical lines have an undefined slope.
• A horizontal line (e.g., the line represented by equation \( y = -2 \)) is always perpendicular to a vertical line.

To identify if two lines are perpendicular without graphing, look at their slopes. If you have one line with a slope \( m_1 \), the line perpendicular to it will have a slope \( m_2 \), where \( m_1 \times m_2 = -1 \). Remember, perpendicularity in terms of horizontal and vertical lines defies the slope product rule since vertical lines do not have an undefined numeric slope!

Horizontal and Vertical Lines

Lines in a coordinate plane can often be categorized as horizontal or vertical, depending on their orientation.

**Horizontal Lines:** These lines stretch from left to right across the plane.

• Their equation is typically \( y = b \), where \( b \) is the y-coordinate of every point on the line.
• Horizontal lines have a slope of zero because they don't rise or fall regardless of how far along the x-axis they go.

**Vertical Lines:** On the other hand, vertical lines rise straight up and down.

• These lines are represented by \( x = a \), where \( a \) is the x-coordinate of every point on the line.
• Vertical lines have undefined slopes because the change in y does not lead to a change in x — they don’t run horizontally at all.

Knowing how to distinguish between these two line types will aid in visualizing and solving geometric problems efficiently.

Equation of a Line

The equation of a line in a two-dimensional cartesian plane can be expressed in different forms, each serving its purpose.

**Slope-Intercept Form:** Likely the most familiar to many, it's expressed as \( y = mx + b \).

• Here, \( m \) represents the slope, showing how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

This form is straightforward and ideal for charting linear relationships. Nonetheless, there are scenarios where a line cannot be expressed in slope-intercept form. For instance, vertical lines like \( x = -5 \) do not have a slope \( m \), and thus, cannot fit into the \( y = mx + b \) model.Different scenarios may call for alternative representations, such as:

• **Point-Slope Form:** Suitable when you know a point and the slope.
• **Standard Form:** Often useful in more complex algebraic manipulations.

Choosing the appropriate form requires understanding the context and objective of the problem.