Problem 107 - College Algebra

Question

Find the equation of the line that passes through the following points: \((a, b)\) and \((a, b+1)\).

Step-by-Step Solution

Answer

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

Step 1: Identify Given Points

We are given two points: \((a, b)\) and \((a, b+1)\). These points have the same x-coordinate \(a\), meaning the line is vertical.

Step 2: Determine Slope of the Line

The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = x_2 = a\), so the formula gives \(m = \frac{b+1 - b}{a - a} = \frac{1}{0}\), which is undefined. This confirms the line is vertical.

Step 3: Write the Equation of the Line

A vertical line through the point \((a, b)\) has the equation \(x = a\). The line does not depend on the y-value differences because vertical lines have an undefined slope.

Key Concepts

Vertical Lines

A vertical line is a straight line that runs up and down. It is parallel to the y-axis. One unique feature of vertical lines is that their x-coordinates are constant. They do not change as y-coordinates vary. For example, the points

• \((a, b)\)
• \((a, b+1)\)

are positioned on a vertical line because they share the same x-coordinate, \(a\). When graphed, vertical lines visually show the concept that x-values remain stable across various y-values. As a result, vertical lines cannot be expressed in the form of \(y = mx + c\) like other linear equations but are simply \(x = a\), where \(a\) is the constant x-value. Recognizing that a line is vertical based on its constant x-coordinate is crucial since it affects how we calculate its characteristics.

Slope Calculation

The slope of a line measures how steep or flat the line is. It is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This represents the change in y for a unit change in x between any two points

• \((x_1, y_1)\)
• \((x_2, y_2)\)

When using this formula, it's essential to know that:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls from left to right.
• A zero slope shows a horizontal line.

In our example, the identical x-coordinates \(a\) for the two points \((a, b)\) and \((a, b+1)\) leads to a division by zero in the slope calculation, a unique scenario to vertical lines.

Undefined Slope

An undefined slope occurs when there is a division by zero in the slope formula. This happens uniquely in vertical lines. When calculating the slope \(m\) between two points where \(x_1 = x_2\), the formula becomes \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_2 - y_1}{0}\]Mathematically, division by zero is undefined, hence the term "undefined slope." This tells us that the line doesn't run horizontally or diagonally but straight up and down. Recognizing an undefined slope quickly indicates a vertical line, which impacts your approach in finding the equation of the line. The line does not fit the usual y = mx + c form, highlighting its unique nature in geometry.