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Chapter 1: Problem 82

Is there a difference between the statements "The slope of a straight line is zero" and "The slope of a straight line does not exist (is not defined)"? Explain your answer.

Short Answer

Expert verified
Yes, there is a difference between the statements. When the slope of a straight line is zero, it means the line is horizontal, with no vertical change between any two points on the line. On the other hand, when the slope of a straight line does not exist (is not defined), it means the line is vertical, and calculating the slope would require dividing by zero, which is undefined. In summary, a zero slope represents a horizontal line, while an undefined slope represents a vertical line.

Step by step solution

01

Understand the concept of slope

The slope of a straight line tells us how steep the line is. It can be positive, negative, or zero, and it describes the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The formula to calculate the slope is given by: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where m is the slope and (x1, y1) and (x2, y2) are two points on the line.
02

Visualize a line with a zero slope

If the slope of a straight line is zero, it means that the line is horizontal (parallel to the x-axis), with no vertical change between any two points on the line. In this case, the rise is zero, and the slope calculation becomes: \(m = \frac{0}{x_2 - x_1} = 0\)
03

Visualize a line with an undefined slope

A slope is considered to be undefined when the vertical change (rise) is not zero and the horizontal change (run) is zero. This would mean that the line is vertical (parallel to the y-axis). For a vertical line, the slope is undefined because the denominator in the slope calculation formula becomes zero, leading to a division by zero, which is undefined: \(m = \frac{y_2 - y_1}{0}\)
04

Compare the two cases

To summarize: - When the slope of a straight line is zero, it means the line is horizontal, with no vertical change between any two points on the line. - When the slope of a straight line does not exist (is not defined), it means the line is vertical, and calculating the slope would require dividing by zero. These two situations are clearly different, as the first involves a horizontal line and the second involves a vertical line.

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Most popular questions from this chapter

Determine whether the points lie on a straight line. $$ A(-1,7), B(2,-2), \text { and } C(5,-9) $$

Find an equation of the line that passes through the point \((1,-2)\) and is perpendicular to the line passing through the points \((-2,-1)\) and \((4,3)\).

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