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Chapter 3: Problem 50

Find the slope of the line containing each given pair of points. If the slope is undefined, state this. $$ (-10,3) \text { and }(-10,4) $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

Identify the coordinates

Label the coordinates of the two given points. Let \((x_1, y_1) = (-10, 3)\) and \((x_2, y_2) = (-10, 4)\).
02

Use the slope formula

Recall the formula for the slope of a line: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. Substituting the given points into the formula yields: \[m = \frac{4 - 3}{-10 - (-10)}\].
03

Calculate the differences

Calculate the numerator and the denominator separately.\[y_2 - y_1 = 4 - 3 = 1\] and \[x_2 - x_1 = -10 - (-10) = -10 + 10 = 0\].
04

Determine the slope

Substitute the differences back into the slope formula: \[m = \frac{1}{0}\]. Since the denominator is 0, the slope is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

coordinates
To understand the slope of a line, the first thing we need to know is the coordinates of the points on that line. Coordinates are written as pairs of numbers \(x, y\), representing their position on a graph. For example, the coordinates \((-10, 3)\) indicate a point located at -10 on the x-axis and 3 on the y-axis. These coordinates are crucial for calculating the slope. By labeling coordinates, such as \((x_1, y_1)\) and \((x_2, y_2)\), we can easily plug them into the slope formula.
slope formula
The slope of a line is a measure of how steep it is. The slope formula is expressed as \[m = \frac{y_2 - y_1}{x_2 - x_1}\].
This formula requires two points, \((x_1, y_1)\) and \((x_2, y_2)\), on the line. Here's how you use it:
  • Subtract the first y-coordinate from the second y-coordinate to find the numerator.
  • Subtract the first x-coordinate from the second x-coordinate to find the denominator.
  • Divide the numerator by the denominator to get the slope, or \(m\).
In our example, we substitute the points \( (-10, 3) \) and \( (-10, 4) \) into the slope formula: \[m = \frac{4 - 3}{-10 - (-10)} = \frac{1}{0}\].
undefined slope
What happens if the denominator in the slope formula is zero? This results in an undefined slope.
  • If the x-coordinates are the same (as in \(-10, 3\) and \(-10, 4\)), the line is vertical.
  • A vertical line means the x-values do not change, so the slope cannot be defined.
  • In our case, we discovered \(x_2 - x_1 = 0\), which led to \[m = \frac{1}{0}\].
Since division by zero is impossible, the slope is considered undefined. Thus, the line passing through points \((-10, 3)\) and \((-10, 4)\) has an undefined slope, indicating it's a vertical line.

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Most popular questions from this chapter

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