Problem 16 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((4,7)\) whose slope is undefined

Step-by-Step Solution

Answer

The equation of the line is \(x = 4\).

Step 1: Identify the Characteristics of the Line

A line with an undefined slope is a vertical line. Vertical lines have equations of the form \(x = a\) where \(a\) is the x-coordinate of the point through which the line passes.

Step 2: Find the x-coordinate

Since the line passes through the point (4, 7), the x-coordinate \(a\) is 4.

Step 3: Write the Equation of the Line

Using the form of the equation for a vertical line \(x = a\), and substituting \(a = 4\), the equation is \(x = 4\).

Key Concepts

undefined slope

In coordinate geometry, the slope of a line quantifies its steepness and direction. However, when we talk about an undefined slope, we are dealing with a special scenario. An undefined slope occurs in vertical lines. This means that for a vertical line, the slope does not have a defined value. Why? Because the change in y-values happens without any change in x-values.

In a vertical line, all points share the same x-coordinate, yet can have any y-coordinate. Therefore, the conventional slope formula, which is \(\frac{Δy}{Δx}\), would require division by zero — something mathematically impossible and hence 'undefined'.

vertical line equation

A vertical line has a distinctive feature in its equation. Unlike other lines, which usually follow the format \(y = mx + b\), a vertical line is represented as \(x = a\). Here, 'a' is the constant x-coordinate for every point on the line.

Let's take an example to make this clearer. If you have a point (4, 7), a vertical line passing through this point will include every point with x=4. So, the equation becomes \(x = 4\). It does not matter what the y value is, as long as the x value remains 4.

coordinate geometry

Let's delve deeper into coordinate geometry. This branch of mathematics essentially allows us to represent algebraic equations through geometric visualization. Coordinates are represented as \((x, y)\) pairs, mapping specific points on a plane.

Things get more interesting with vertical lines. These lines, represented by equations like \(x = 4\), correspond to all points whose x-coordinate is fixed at a certain value, irrespective of their y-coordinate. This geometric representation helps in visualizing how lines and curves behave in a two-dimensional space.

In summary, mastering the concepts of undefined slopes, vertical line equations, and coordinate geometry helps build a solid foundation for solving diverse problems in mathematics.