Problem 43 - Brief 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 Points

The points given are (1, -1) and (5, -1).

Step 2: Determine the Slope (m)

The formula for the slope, \(m\), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2-y_1}{x_2-x_1}\). Substitute the values from the points: \(m = \frac{-1 - (-1)}{5-1} = \frac{0}{4} = 0\).

Step 3: Write the Equation Using Slope-Intercept Form

Since the slope \(m\) is 0, the line is horizontal. In the slope-intercept form \(y = mx + b\), \(y\) will be a constant \(b\). Here, since \(y = -1\) for both points, \(b = -1\). Therefore, the equation is \(y = 0 \cdot x - 1\) or simply \(y = -1\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a straight line. It is written as \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( m \) is the slope of the line.
• \( x \) is the independent variable.
• \( b \) is the y-intercept of the line, which is where the line crosses the y-axis.

This form is especially useful because it gives immediate insights into the slope and position of the line
on a graph. The slope \( m \) indicates the steepness and direction of the line, while \( b \) allows us to know where the line intersects the y-axis. In this exercise, since the slope \( m = 0 \), the line has no vertical change, thus it is horizontal.

Horizontal Line Equation

A horizontal line is a line where all points share the same y-coordinate. Therefore, the equation of a horizontal line can be represented as \( y = b \), where \( b \) is the constant y-value for that line.

• In this exercise, both points (1, -1) and (5, -1) have the same y-value of -1.
• This indicates that the line is horizontal and unchanging as x varies.

Hence, the equation of the line becomes \( y = -1 \). Horizontal lines have a slope of zero, which means there is no vertical movement as x changes. Therefore, in any equation of a horizontal line written in slope-intercept form \( y = mx + b \), \( m \) equals zero.

Slope Calculation

Calculating the slope is key when you have two points and want to determine the angle and direction of the line connecting them. The slope \( m \) can be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

• The numerator \( y_2 - y_1 \) represents the vertical change (rise).
• The denominator \( x_2 - x_1 \) indicates the horizontal change (run).

For the points (1, -1) and (5, -1) given in the exercise, the slope is \( m = \frac{-1 - (-1)}{5 - 1} = 0 \).
This shows no vertical change as we move between the points, confirming the line is horizontal.