Problem 39 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The equation of the line is \(x = 1.5\).

Step 1: Understanding the Problem

We need to find the equation of a line that is vertical and passes through the point (1.5, -4). Vertical lines have undefined slope and cannot be expressed in the form \(y = mx + b\). Instead, vertical lines are expressed in the form \(x = a\), where \(a\) is the x-coordinate that the line passes through.

Step 2: Identifying the x-coordinate

The given point is (1.5, -4). Since the line is vertical, the equation will be based on the x-coordinate of the given point. For this problem, the x-coordinate is 1.5.

Step 3: Writing the Equation

Since the line is vertical and passes through the point (1.5, -4), the equation of the line is \(x = 1.5\). Vertical lines only have equations of the form \(x = a\) and do not have a y-variable in their linear equation form.

Key Concepts

Vertical Line Equation

A vertical line is a unique type of straight line in geometry. Unlike most lines, vertical lines have no slope, meaning they are perpendicular to the x-axis and go straight up and down. Because of this, they cannot be represented by the usual linear equation form \(y = mx + b\). This form doesn't fit because it implies a slope \(m\), and vertical lines do not have a defined slope. Instead, vertical lines are expressed as \(x = a\), where \(a\) is a constant corresponding to the x-coordinate each point on the line shares.
For example, if a line is vertical through the point (1.5, -4), every point on this line has an x-coordinate of 1.5. Thus, the equation for this line is \(x = 1.5\). This shows that the line runs parallel to the y-axis, cutting through every height where x remains 1.5. In other words, no matter where you are vertically on the line, the x-value stays constant.

Linear Equation Form

The linear equation form \(y = mx + b\) is commonly used to express straight lines in the coordinate plane. It is called the "slope-intercept form" because its parameters directly reflect the line's slope and y-intercept.

• Here, \(m\) represents the slope of the line, indicating its steepness and direction.
• The variable \(b\) denotes the y-intercept, the point where the line crosses the y-axis.

The standard linear form is excellent for lines that slope upwards or downwards. However, it doesn't apply to vertical lines, as vertical lines have no slope and run parallel to the y-axis. Therefore, for vertical lines, their equations take the form \(x = a\) where \(a\) is the value that describes the shared x-coordinate.

Slope of a Line

The slope of a line, often represented by \(m\), is a measure of its steepness and direction. In the slope-intercept form \(y = mx + b\), the slope is crucial for determining how the line moves across the coordinate plane. If you think of driving a car up a hill, the slope tells you how steep the hill is:

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• A zero slope means the line is flat, like a road with no incline.

However, a vertical line is something different. It has an undefined slope because its rise/run calculation involves division by zero (as the "run" part is zero). The undefined slope tells us the line is vertical, with an equation formatted as \(x = a\). This concept clarifies why we can't express vertical lines with \(y = mx + b\), as the slope \(m\) cannot be determined or applied.