Problem 53 - 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. Perpendicular to \(x=3,\) passing through \((1,2)\)

Step-by-Step Solution

Answer

The equation is \(y = 2\).

Step 1: Analyze the Given Line

The line given by the equation \(x = 3\) is a vertical line. Vertical lines have an undefined slope.

Step 2: Determine the Slope of the Perpendicular Line

A line that is perpendicular to a vertical line (undefined slope) is a horizontal line. Horizontal lines have a slope of \(0\).

Step 3: Use Point-Slope Form to Establish the Equation

Knowing the slope is \(0\) and it passes through the point \((1, 2)\), use the equation of a horizontal line \(y = b\), where \(b\) is the y-coordinate of the given point. Thus, the equation is \(y = 2\).

Step 4: Confirm the Slope-Intercept Form

The resulting equation \(y = 2\) is already in slope-intercept form \(y = mx + b\), where \(m = 0\) and \(b = 2\).

Key Concepts

Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle (90 degrees). This is an important concept when determining the relationship between two lines in a coordinate plane. If you know the slope of one line, you can find the slope of a perpendicular line using a simple rule: the product of their slopes should be \(-1\). However, things change a bit when dealing with vertical and horizontal lines.

• Vertical lines have an undefined slope.
• Horizontal lines have a slope of zero.

Thus, a line that is perpendicular to a vertical line must be a horizontal line, and vice versa, because these types of lines naturally create right angles where they meet.

Horizontal Line Equation

A horizontal line is one where all points have the same y-coordinate, no matter what the x-value is. This can be represented by the equation \(y = b\), where \(b\) is the constant y-coordinate for the entire line. When a line is horizontal, its slope is zero because there is no vertical change as you move along it.

• For example, the line \(y = 2\) passes through every point where the y-coordinate is 2.
• A horizontal line does not slant in any direction.
• It provides a perfect illustration of zero slope as the rise (change in y) stays constant while the run (change in x) varies.

This makes horizontal lines simple yet crucial when identifying perpendicular relationships in coordinate geometry.

Point-Slope Form

The point-slope form is a way to write the equation of a line if you know a point on the line and its slope. This form is written as \(y - y_1 = m(x - x_1)\), where:

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope of the line.

In scenarios where a horizontal line's slope is identified as zero, the equation simplifies significantly. Since the slope \(m = 0\), the equation becomes \(y - y_1 = 0\), which further simplifies to \(y = y_1\). This example demonstrates how point-slope form aids in easily transitioning to the equation of a horizontal line. The elegance of the point-slope form is its flexibility, making it straightforward to derive line equations from minimal initial information.