Problem 37 - Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Horizontal and passing through the point (1.5,-4)

Step-by-Step Solution

Answer

The equation is \(y = -4\).

Step 1: Understanding Horizontal Lines

A horizontal line in the coordinate plane is a line where all points on the line have the same y-coordinate. This means that the equation of a horizontal line is in the form \(y = c\), where \(c\) is the constant y-coordinate for all points on the line.

Step 2: Identify the Y-coordinate

The given point through which the horizontal line passes is \((1.5, -4)\). Since the line is horizontal, the y-coordinate for all points on the line will be \(-4\).

Step 3: Formulate the Equation

Using the y-coordinate identified in step 2, the equation of the line is \(y = -4\). This represents a horizontal line where every point on the line has a y-coordinate of \(-4\).

Key Concepts

Horizontal Line

A horizontal line is one that runs left to right across the coordinate plane, parallel to the x-axis. One key characteristic of a horizontal line is that all its points have the same y-coordinate, no matter what x-values they take.
This is why the equation of any horizontal line is given by \(y = c\), where \(c\) is a constant. In the case of the exercise, since the horizontal line passes through the point \((1.5, -4)\), our constant \(c\) becomes \(-4\), resulting in the equation \(y = -4\).
Horizontal lines are special because they do not have a slope in the traditional sense. Their 'slope' is considered 0, since there is no vertical change as you move along the line. This means that no matter how far you travel horizontally, the vertical position remains constant.

Coordinate Plane

The coordinate plane, also known as a Cartesian plane, is a two-dimensional plane formed by the intersection of a vertical line, called the y-axis, and a horizontal line, called the x-axis. These axes allow us to uniquely define every point on the plane by a pair of numbers.
Each point in the coordinate plane can be described using an ordered pair \((x, y)\), where \(x\) represents the point's horizontal position and \(y\) represents its vertical position. In the exercise, the point \((1.5, -4)\) is located by moving 1.5 units to the right on the x-axis and 4 units down on the y-axis.
Understanding the coordinate plane is fundamental because it provides a visual way to understand relationships such as the equation of a line. By plotting the points, one can easily see how a line behaves and interacts with different parts of the plane.

Y-coordinate

The y-coordinate of a point in the coordinate plane tells us how far the point is in the vertical direction from the x-axis. It is the second number in an ordered pair \((x, y)\). For a horizontal line, which runs parallel to the x-axis, the y-coordinate remains constant across all the points on the line.
In the given exercise, the point \((1.5, -4)\) shows that the y-coordinate is \(-4\). This means the entire line passes through the y-coordinate of \(-4\) no matter what the x-value is.
Understanding the role of y-coordinates in forming equations of lines is crucial, especially when identifying horizontal lines. Keeping the y-coordinate constant is what defines the line's horizontality. So, in the equation \(y = -4\), \(-4\) essentially "cements" the line's vertical position on the plane.