Question
Why does the slope formula not apply to vertical lines?
Answers
GPT-4o mini
The slope formula is defined as the change in \(y\) divided by the change in \(x\), mathematically expressed as:
\[
\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, this formula calculates the slope by measuring how much \(y\) changes for a given change in \(x\).
In the case of a vertical line, all points on the line have the same \(x\)-coordinate. For instance, consider the line represented by \(x = a\), where \(a\) is a constant. If you select two points on this vertical line, say \((a, y_1)\) and \((a, y_2)\), you will notice that:
\[
x_1 = a \quad \text{and} \quad x_2 = a
\]
Thus, when you substitute these coordinates into the slope formula, you get:
\[
m = \frac{y_2 - y_1}{a - a} = \frac{y_2 - y_1}{0}
\]
Since you cannot divide by zero, the slope formula fails to produce a valid numerical value for vertical lines.
Therefore, we say that vertical lines have an **undefined slope** because the concept of slope requires a non-zero change in \(x\) to calculate how \(y\) changes with respect to \(x\). In essence, a vertical line represents an infinite vertical rise (change in \(y\)) for no horizontal run (change in \(x\)), which mathematically results in an undefined slope.
\[
\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, this formula calculates the slope by measuring how much \(y\) changes for a given change in \(x\).
In the case of a vertical line, all points on the line have the same \(x\)-coordinate. For instance, consider the line represented by \(x = a\), where \(a\) is a constant. If you select two points on this vertical line, say \((a, y_1)\) and \((a, y_2)\), you will notice that:
\[
x_1 = a \quad \text{and} \quad x_2 = a
\]
Thus, when you substitute these coordinates into the slope formula, you get:
\[
m = \frac{y_2 - y_1}{a - a} = \frac{y_2 - y_1}{0}
\]
Since you cannot divide by zero, the slope formula fails to produce a valid numerical value for vertical lines.
Therefore, we say that vertical lines have an **undefined slope** because the concept of slope requires a non-zero change in \(x\) to calculate how \(y\) changes with respect to \(x\). In essence, a vertical line represents an infinite vertical rise (change in \(y\)) for no horizontal run (change in \(x\)), which mathematically results in an undefined slope.