Problem 44 - 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 (2,0) and (2,-4)

Step-by-Step Solution

Answer

The line is vertical with the equation \(x = 2\).

Step 1: Determine the type of line

Since the line passes through the points (2,0) and (2,-4), both x-coordinates are the same, which indicates a vertical line.

Step 2: Vertical line equation

The equation of a vertical line passing through a given x-coordinate is simply given in the form: \(x = a\), where \(a\) is the x-coordinate common to all points on the line.

Step 3: Find the x-coordinate

The x-coordinate common to both points (2, 0) and (2, -4) is 2.

Step 4: Write the equation of the line

Thus, the equation of the line is \(x = 2\). As this is a vertical line, it cannot be expressed in the form \(y = mx + b\).

Key Concepts

Equation of a Line

At its core, an equation of a line represents the relationship between all the points that lie on that line. In many scenarios, this equation is given as the slope-intercept form, which is written as \(y = mx + b\). This form is very helpful because it explicitly shows the slope \(m\) and the y-intercept \(b\), which are key characteristics of non-vertical and non-horizontal lines. The slope \(m\) measures how steep the line is, while the y-intercept \(b\) indicates where the line crosses the y-axis.
Lines can also be expressed through other forms such as the standard form \(Ax + By = C\) or the point-slope form \(y - y_1 = m(x - x_1)\). Choosing the right form often depends on the specific nature or positioning of the line being examined.

Vertical Lines

Vertical lines are a unique and special class of lines. When a line is vertical, it means it has an undefined slope. This is because a vertical line moves straight up and down, parallel to the y-axis, and doesn't rise or run in a traditional sense. The equation of a vertical line is given by \(x = a\), where \(a\) is the x-coordinate shared by all points on the line. This is due to the fact that in a vertical line, every point has the same x-value.

• Example: For points (2,0) and (2,-4), the vertical line equation is \(x = 2\).
• Vertical lines will never intersect the y-axis, hence they do not have a y-intercept.

In summary, when both x-coordinates of two points are equal, it indicates the presence of a vertical line.

Understanding Line Types

Knowing the different types of lines is crucial in graphing and mathematical problem-solving.

Lines generally come in three basic forms:

• **Horizontal Lines:** These lines are parallel to the x-axis and have a slope of 0. They are written as \(y = b\), where \(b\) is the constant y-coordinate of all points on the line.
• **Vertical Lines:** As we've discussed, vertical lines run parallel to the y-axis with an undefined slope. They are expressed as \(x = a\).
• **Oblique Lines (non-horizontal or vertical):** These lines slope up or down and are typically expressed as \(y = mx + b\). Their slope \(m\) defines the angle of incline relative to the horizontal plane.

Understanding line types helps in identifying the right equations to use in geometry and aids in visualizing the relationships between points on a coordinate plane. Each type of line tells you different things about its movement and how it interacts with the axes in a graph.