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Chapter 1: Problem 14

Find the slope (if it is defined) of the line determined by each pair of points. \((6,-4)\) and \((6,-3)\)

Short Answer

Expert verified
The slope is undefined; the line is vertical.

Step by step solution

01

Identify Points

The two points given are \((6, -4)\) and \((6, -3)\). We need to calculate the slope of the line passing through these points.
02

Slope Formula

The formula for the slope \(m\) of the line determined by two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
03

Substitute Values

Substitute \((6, -4)\) and \((6, -3)\) into the slope formula: \(m = \frac{-3 - (-4)}{6 - 6}\).
04

Calculate Numerator

Calculate the numerator: \(-3 - (-4) = -3 + 4 = 1\).
05

Calculate Denominator

Calculate the denominator: \(6 - 6 = 0\).
06

Determine Slope Status

Since the denominator is 0, the slope is undefined. A division by zero indicates the line is vertical.

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Key Concepts

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

Line Through Points
Understanding how to determine a line through given points is a fundamental concept in geometry. Given any two points in a coordinate plane, a line can be drawn to pass through them. This line will have certain properties, namely its slope which describes its steepness and direction.
For the pair of points
  • The first point \( (6, -4) \)
  • The second point \( (6, -3) \)

these coordinates allow us to determine the type of line that connects them. To analyze this line, it helps to first recognize whether the x-coordinates (or y-coordinates) are the same. In this case, both points have the same x-coordinate (6), which suggests the line may be vertical, a unique case that will affect how we calculate and interpret the slope.
Slope of a Line
The slope of a line is a numerical value that describes the line's direction and steepness calculated from two points. It's typically denoted by the letter \( m \) and given by the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\] Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points through which the line passes. The slope tells us:
  • How tilted or flat a line is.
  • Whether it rises or falls as it moves from left to right.

If the slope is positive, the line rises; if negative, the line falls. If the slope equals 0, the line is horizontal. In the example of points \( (6, -4) \) and \( (6, -3) \), substituting them into the slope formula reveals something crucial about the type of line that these points define.
Vertical Line
A vertical line is a line in which all points share the same x-coordinate, causing a unique condition for slope calculations. In our example:
  • Both points have the x-coordinate 6.
  • The slope calculation results in a denominator of 0: \(x_2 - x_1 = 6 - 6 = 0\).

This results in the slope being undefined because division by zero is mathematically impossible. A vertical line poses this scenario because it implies a perfectly up-and-down direction that doesn't lean left or right. This peculiarity of the vertical line is crucial to recognize when calculating slopes, indicating a special type of line always with an undefined slope.

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