Problem 52 - College Algebra with Modeling and Visualization

Question

Find an equation of the line satisfying the conditions. Horizontal, passing through \((1.95,10.7)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 10.7 \).

Step 1: Identify the Nature of the Line

Since the line is described as a horizontal line, recall that a horizontal line has a constant y-value for all x. This means its slope is 0.

Step 2: Determine the Equation of a Horizontal Line

A horizontal line at any point has the equation of the form \( y = c \), where \( c \) is the constant y-value the line passes through.

Step 3: Substitute the Given Point into the Equation

Given that the line passes through the point \((1.95, 10.7)\), substitute \(10.7\) into \(c\). Thus, the equation of the line becomes \( y = 10.7 \).

Key Concepts

The Nature of a Horizontal Line

A horizontal line in coordinate geometry is a line that runs straight from left to right. This type of line has a unique property: it joins all points that have the same y-coordinate. This means as you move along the line, even though the x-coordinates might change, the y-coordinate remains constant. Therefore, if you have any horizontal line, regardless of where it is located on the plane, the y-value across all of its points will be uniform.

This constant y-value is what defines the horizontal line's equation. For every horizontal line, the equation takes the form of \( y = c \), where \( c \) is the specific y-coordinate of every point on the line. This simplicity in definition is helpful in quickly identifying or calculating equations of horizontal lines in geometrical problems.

Understanding the Slope of a Line: Zero Slope

When talking about the slope of a line, we're essentially discussing how steep the line is. This concept comes into play when you want to understand how much y changes as x changes. For horizontal lines, the slope is quite special. - Horizontal lines are perfectly flat.- The slope of a horizontal line is \( 0 \).
The reason for this is simple: since the line is horizontal, there is no rise (or change in the y-value) when moving along the line; x might change, but y doesn't. This zero change in the y-value indicates that the numerator in the slope formula, which is \( \Delta y \), is zero. The slope formula is calculated as: \[m = \frac{\Delta y}{\Delta x}\]Since \( \Delta y = 0 \) for horizontal lines, the slope \( m \) becomes \( 0 \), illustrating the flat nature of the line.

Coordinate Geometry and Line Equations

Coordinate geometry, also known as analytic geometry, is a way of studying geometry using a coordinate system. This branch of mathematics enables us to pinpoint the locations and paths of geometric figures using algebraic equations.

When it comes to line equations, coordinate geometry offers a systematic approach. For example:- Vertical lines have equations of the type \( x = c \).- Horizontal lines are described by equations like \( y = c \), indicating a constant y-value.
Through equations like these, coordinate geometry helps in visualizing how lines and shapes behave within the plane. Specifically, horizontal lines hold a special place because they form perfect parallels to the x-axis, maintaining one fixed y-value throughout their length. This makes them easy to work with and identify in geometric contexts.