Problem 26 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Vertical line passing through \((2,0)\)

Step-by-Step Solution

Answer

x = 2

Step 1: Identify key properties of a vertical line

A vertical line has an undefined slope and all points on the line share the same x-coordinate. In this case, the line must contain the point \(2,0\), meaning every point on the line will have an x-coordinate of 2.

Step 2: Write the equation of the vertical line

Since the vertical line passes through the point \(2,0\) and has the same x-coordinate for all points, the equation of the line is x = 2.

Key Concepts

undefined slope

Vertical lines are unique in geometry. Unlike other lines, a vertical line has an undefined slope. This means that no matter how steep a line may be, if it is vertical, its slope calculation will be impossible.
To understand why, think about the slope formula:
\(m = \frac{\Delta y}{\Delta x}\).
In a vertical line, \(\Delta x \) is always zero because \(x\) does not change. And dividing by zero is undefined, leading to an undefined slope.
Therefore, if a line is vertical, we can't describe its steepness using a numerical value.

x-coordinate

A key feature of vertical lines is that all points on the line share the same x-coordinate.
For instance, if a vertical line passes through the point \(2,0\), every point on this line will have an x-coordinate of \2\. This is because the line stretches infinitely in the up and down directions but does not move left or right.
In our exercise, since the line passes through \(2,0\), the x-coordinate for every point on the line is \2\. Therefore, we can represent this as the simple equation \(x = 2\). This equation tells us that no matter what the y-value is, the x-value will always be \2\.

algebra

Understanding vertical lines also involves basic algebra principles. Algebra helps us translate words into equations. When given the points a line must pass through, algebra allows us to form the line’s equation.
Here, knowing a vertical line has an undefined slope and a fixed x-coordinate, we can write its equation simply and clearly. For example, the given point is \(2,0\), leading us to \(x = 2\).
Algebra helps simplify these steps by offering a straightforward way to express geometric concepts in an equation form, helping us solve and understand more complex problems with ease.