Problem 43 - Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Passing through the points (1,-1) and (5,-1)

Step-by-Step Solution

Answer

The equation of the line is \(y = -1\).

Step 1: Identify the Components of the Line Equation

Given two points (1, -1) and (5, -1), we need to find the equation of the line passing through these points. The general form of a line is given by the equation \(y = mx + b\). Our task is to determine the slope \(m\) and the y-intercept \(b\).

Step 2: Calculate the Slope of the Line

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points (1, -1) and (5, -1), we have: \[ m = \frac{-1 - (-1)}{5 - 1} = \frac{0}{4} = 0 \]Thus, the slope \(m = 0\).

Step 3: Determine the Y-intercept

Since the slope \(m = 0\), the line is horizontal. To find the y-intercept \(b\), observe that the y-value of the points is constant at -1. Therefore, the equation of the line is \(y = -1\).

Step 4: Write the Equation of the Line

Using the slope and y-intercept, the equation of the line through the points (1, -1) and (5, -1) is simply a horizontal line: \[ y = -1 \]

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It tells us how much the line inclines or declines as it moves from left to right. For any line passing through two points, the slope \( m \) is calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The numerator \( (y_2 - y_1) \) represents the vertical change between the two points, while the denominator \( (x_2 - x_1) \) represents the horizontal change. In simple terms, this formula helps compare the rise to the run of the line.
For example, in the current exercise, the given points are \( (1, -1) \) and \( (5, -1) \).
Calculating the slope:

• The difference in \( y \)-values is zero: \((-1) - (-1) = 0\)
• The difference in \( x \)-values is positive: \( 5 - 1 = 4 \)

Thus, the slope \( m \) becomes zero: \( \frac{0}{4} = 0 \). A slope of zero indicates that the line does not rise or fall as it moves horizontally. This conclusion leads us to understand that when the slope is zero, the line is horizontal.

Y-Intercept

The y-intercept is the point at which a line crosses the y-axis. This is significant because it describes where the line is located vertically, especially when \( x = 0 \). In the equation of a line \( y = mx + b \), the \( b \) represents the y-intercept. It serves as a starting point for graphing a line.
For horizontal lines, the slope \( m \) equals zero. This implies that the line neither rises nor falls, remaining flat. As a result, the y-value (or y-intercept) remains constant across all points on the line. In our example, the line's equation is \( y = -1 \), indicating the y-intercept is \(-1\).
This means that no matter what \( x \)-value you have, \( y \) will always equal \(-1\). Understanding the y-intercept is crucial, as it simplifies interpreting the line's position on the graph, especially for horizontal lines.

Horizontal Line

Horizontal lines are special because they have a uniform height across their length. In mathematical terms, a horizontal line has a slope of zero. This is because there is no change in the y-value as the x-value changes. Therefore, every point on a horizontal line will have the same y-coordinate.
In our problem, since the slope is calculated to be zero (as seen with the points \( (1, -1) \) and \( (5, -1) \)), the line remains flat. The equation of this horizontal line simplifies to \( y = -1 \).
This equation tells us that the line remains at y-coordinate \(-1\) for all x-coordinates. Such lines appear as flat, unchanging horizontals in graph representations. They can be vital for illustrating concepts in algebra relating to constant functions where output does not vary.