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Chapter 1: Problem 13

Find the slope (if it is defined) of the line determined by each pair of points. \((2,-1)\) and \((2,5)\)

Short Answer

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

Step by step solution

01

Understanding the Formula

The formula for calculating the slope \( m \) of a line given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is defined as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
02

Substituting the Given Points

Insert the given points \( (2, -1) \) as \( (x_1, y_1) \) and \( (2, 5) \) as \( (x_2, y_2) \) into the slope formula. This gives: \[ m = \frac{5 - (-1)}{2 - 2} \].
03

Calculating the Slope

Compute the numerator and the denominator. For the numerator: \( 5 - (-1) = 6 \). The denominator is: \( 2 - 2 = 0 \). Thus, the equation becomes \[ m = \frac{6}{0} \].
04

Determining the Result

Since the denominator is zero, the slope is undefined. This means the line is vertical.

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Key Concepts

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

vertical lines
Vertical lines are a special type of line in coordinate geometry. They run straight up and down on a graph.
This means that, visually, they do not tilt to the left or right, unlike most other lines.
  • A vertical line has an equation in the form of \( x = a \), where \( a \) is a constant.
  • All points on this line have the same \( x \)-coordinate, while the \( y \)-coordinates can differ.
  • The line in our exercise, passing through \((2, -1)\) and \((2, 5)\), has the equation \( x = 2 \).
Visualize vertical lines as if you're looking at a ladder that you can climb up or down, but you cannot step to the left or right.Vertical lines are key markers on the graph that help define positions where \( x \) does not change.
undefined slope
An undefined slope occurs when you attempt to calculate the slope of a vertical line.
Remember the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • If the \( x \)-coordinates of the two points are the same, the denominator becomes zero.
  • As seen in the exercise, the calculation results in \( \frac{6}{0} \).
  • Any division by zero in mathematics is undefined, hence `undefined slope`.
This concept simply means that for vertical lines, no numerical value can effectively represent the "steepness" or "tilt" of the line.
Conceptually, trying to tilt a ladder vertically does not make sense, because it鈥檚 already perfectly upright.
coordinate geometry
Coordinate geometry, often known as analytic geometry, is a branch of mathematics that uses algebra to describe and explore geometric figures.
It allows us to represent geometric shapes numerically and derive their properties using formulas and equations.
  • Points are defined by pairs of numbers, \((x, y)\), that indicate positions on a plane.
  • Lines are equations that describe a set of points arranged in a straight path with a constant slope between any two points.
  • In the context of lines, you'll often delve into concepts such as slopes, which indicate steepness.
For the points \((2, -1)\) and \((2, 5)\), coordinate geometry allows us to identify that they form a vertical line.
Using \((x_1, y_1)\) and \((x_2, y_2)\) coordinates helps to calculate the slope and provides a visual understanding of geometric relations.

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