Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Slope undefined; containing the point (2,4)

Short answer

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

Step-by-step solution

Understand the problem

The problem specifies that the line has an undefined slope and passes through the point (2, 4).

Determine the nature of the line

A line with an undefined slope is a vertical line. This means its equation will be of the form \( x = a \), where \( a \) is the x-coordinate of any point the line passes through.

Use the given point

Since the point (2, 4) lies on the line, the x-coordinate of every point on this vertical line must be 2.

Write the equation

Thus, the equation of the line is \( x = 2 \).

Key concepts

slope

Understanding the concept of slope is essential in coordinate geometry. Slope indicates the steepness and direction of a line. It's calculated as the ratio of the vertical change (\triangle y) to the horizontal change (\triangle x) between two points on the line. Mathematically, slope (m) is expressed as:
\(m = \frac{\triangle y}{\triangle x}\)
A positive slope means the line rises from left to right, while a negative slope means it falls. Horizontal lines have a slope of zero because there's no vertical change.
Contrarily, vertical lines have an undefined slope since there is no horizontal change, a subject we'll dive into next.

vertical line

In coordinate geometry, a vertical line is one that runs up and down, parallel to the y-axis. Vertical lines are unique because they do not have a slope in the traditional sense.
Instead of following the standard slope formula, the slope of a vertical line is considered undefined.
The equation of a vertical line takes a simple form:
\( x = a \)
where \(a\) is the x-coordinate of any point the line passes through.
For example, in the problem given, the vertical line must pass through the point (2, 4), meaning the equation of the vertical line would be \( x = 2 \).

coordinate geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to describe the position of points, lines, and shapes in space. It uses a coordinate system to express geometric properties and relationships.
In this system, any point on a plane is described by an ordered pair (x, y).
Lines and curves can be represented by equations.
For example, the equation \( y = mx + c \) is a common way to describe a line, where \(m\) is the slope and \(c\) is the y-intercept.
When dealing with vertical lines, the equation simplifies to \( x = a \), illustrating how coordinates help define the shape's position accurately and succinctly.

undefined slope

The concept of an undefined slope can seem tricky at first. To understand it, think about what happens when you try to find the slope of a vertical line.
A slope is generally found using the formula: \( m = \frac{\triangle y}{\triangle x} \).
For a vertical line, \( \triangle x = 0 \) because there's no horizontal change.
Mathematically, dividing by zero is undefined.
Therefore, we say the slope of a vertical line is undefined.
When you encounter a problem stating a line has an 'undefined slope,' you immediately know it's vertical and its equation will be in the form \( x = a \). This property simplifies solving many geometry problems, like our given example.