Undefined Slope of a Line | How to Identify and Calculate It

QUESTION: What is the slope of a line that passes through points (-2, 7) and (-2, 9)?

SOLUTION:

When we examine the points (- 2, 7) and (- 2, 9), we notice that both share the same x-coordinate (-2). This shared x-coordinate tells us that the line passing through these points is vertical, parallel to the y-axis. A vertical line has an undefined slope, which means the slope doesn’t exist as a regular number, and in some cases, we say the slope is “infinite.”

Let’s explore how to determine this using the slope formula.

Slope Calculation Using the Formula:

The slope (m) between two points, $P(x_1,\;y_1)$ and $Q(x_2,\;y_2)$ , is calculated using:

Slope = m = $\frac{change\;in\;y}{change\;in\;x}\;=\;\frac{y_2\;-\;y_1}{x_2\;-\;x_1}$

Step-by-Step Solution:

Step 1: Identify the Points

We have two points:

Point $P(x_1,\;y_1)$ = (- 2, 7) where x₁ = -2 and y₁ = 7
Point $Q(x_2,\;y_2)$ = (- 2, 9) where x₂ = -2 and y₂ = 9

Step 2: Calculate the Change in y

y₂ – y₁ = 9 – 7 = 2

Step 3: Calculate the Change in x

x₂ – x₁ = – 2 – (- 2) ⇒ – 2 + 2 = 0

Step 4: Substitute into the Slope Formula

Slope = m = $\frac{y_2\;-\;y_1}{x_2\;-\;x_1}=\frac{2}{0}$

Since dividing by zero is undefined, the slope is therefore undefined.

Graph

To visualize this, plot the points (-2, 7) and (-2, 9). Draw a line that passes through both points, forming a vertical line parallel to the y-axis. This vertical orientation confirms that the line has an undefined slope, as it doesn’t “rise” or “run” horizontally in a way that can be measured with a regular number.

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