Problem 39 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The equation is \(x = 1.5\).

Step 1: Identify the Type of Line

A vertical line has an undefined slope, and its equation cannot be expressed in the form \(y = mx + b\). Instead, its equation is determined solely by its x-coordinate.

Step 2: Write the Equation of the Line

For a vertical line passing through the point \((1.5, -4)\), the x-coordinate is always 1.5, regardless of the y-coordinate. Therefore, the equation of the line is \(x = 1.5\).

Key Concepts

Vertical Line

A vertical line is a type of line that goes straight up and down, like the sides of a tall building. What makes a vertical line unique is that it does not lean towards the left or right.
Instead, it maintains a constant "x" value for any point on the line. This means that as you move up or down along the line, the "x-coordinate" remains the same. In contrast to other lines that might move diagonally across a graph, vertical lines are easy to spot due to their distinct path that echoes a perfect vertical stance. Graphically, they parallel the y-axis of the graph.

Undefined Slope

When we talk about the slope of a line, we're referring to its steepness or incline. The slope is traditionally calculated as the 'rise over run', or the change in y divided by the change in x between any two points. However, with a vertical line, this formula runs into trouble. Since all points on a vertical line have the same "x" value, there is no change in x — that is, the "run" is zero. Dividing by zero is undefined in mathematics, and thus the slope of a vertical line is termed an "undefined slope". This contrasts with horizontal lines, which have a slope of zero since the "rise" is zero, resulting in a flat line.

x-coordinate

The x-coordinate in a graph represents how far a point is from the y-axis, moving horizontally. It's an integral part of coordinates given as (x, y), where x tells us the horizontal position of a point. For any line, but especially for a vertical line, the x-coordinate is crucial in determining its equation. Because a vertical line does not tilt from side to side, each point on it shares the same x-coordinate.
For example, in the vertical line passing through the point (1.5, -4), every point on this line has an x-coordinate of 1.5. Hence, the equation of the line simply becomes \(x = 1.5\). This illustrates how the x-coordinate exclusively determines the equation of a vertical line.