To determine the slope of the line that passes through the points [tex]\((7,10)\)[/tex] and [tex]\((7,20)\)[/tex], we will follow these steps:
1. Identify the coordinates: We have two points [tex]\((x_1, y_1) = (7, 10)\)[/tex] and [tex]\((x_2, y_2) = (7, 20)\)[/tex].
2. Calculate the difference in x-coordinates and y-coordinates:
- [tex]\(dx = x_2 - x_1 = 7 - 7 = 0\)[/tex]
- [tex]\(dy = y_2 - y_1 = 20 - 10 = 10\)[/tex]
3. Interpret the differences: The formula for calculating the slope [tex]\(m\)[/tex] of a line passing through two points is [tex]\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{dy}{dx}\)[/tex].
4. Substitute the differences into the slope formula:
- [tex]\(m = \frac{10}{0}\)[/tex]
5. Analyze the slope formula: Division by zero is undefined in mathematics. Since [tex]\(dx = 0\)[/tex], the expression [tex]\(\frac{10}{0}\)[/tex] is undefined. This condition indicates that the slope of the line is undefined.
6. Understand the type of line: When the x-coordinates of both points are the same (here [tex]\(x = 7\)[/tex] for both points), the line is vertical. Vertical lines have no slope because the concept of slope (rise over run) doesn’t apply as there is no horizontal movement (run).
7. Conclusion: The line has no slope because [tex]\(x_2 - x_1\)[/tex] is zero, and the denominator of a fraction in the slope formula [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex] cannot be zero.
Given the four options, the correct statement is:
[tex]\[ \boxed{\text{d}} \][/tex]
"It has no slope because [tex]\(x_2 - x_1\)[/tex] in the formula [tex]\(m = \frac{y_2 - y_1}{x_2 - x_1}\)[/tex] is zero, and the denominator of a fraction cannot be zero."
