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Chapter 1: Problem 18

Find the slope of the line containing each pair of points. $$(-7,2),(-7,-6)$$

Short Answer

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

Step by step solution

01

- Understand the Slope Formula

The slope of a line through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \( \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \).
02

- Identify Points

Identify the coordinates of the given points: \( x_1 = -7, y_1 = 2 \) and \( x_2 = -7, y_2 = -6 \).
03

- Substitute Coordinates into Slope Formula

Substitute the coordinates into the slope formula: \[ \text{slope} = \frac{-6 - 2}{-7 - (-7)} \]
04

- Simplify the Expression

Simplify the expression: \[ \text{slope} = \frac{-8}{0} \].
05

- Interpret the Result

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

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Key Concepts

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

slope formula
The slope of a line tells us how steep it is and in which direction it goes. We use the **slope formula** to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\). The formula is: \[\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \].

This simple formula takes the difference in the y-coordinates (vertical change) and divides it by the difference in the x-coordinates (horizontal change).
Here is a brief step-by-step process to use the formula:
  • Identify the coordinates of your two points \(x_1, y_1\) and \(x_2, y_2\).
  • Plug these coordinates into the formula.
  • Subtract the y-coordinates \((y_2 - y_1)\) and then subtract the x-coordinates \((x_2 - x_1)\).
  • Simplify the fraction to get the slope.

Using the slope formula can give you positive, negative, zero, or undefined slopes based on the coordinates of your points.
undefined slope
An **undefined slope** occurs when the line you are analyzing is vertical.

In mathematics, any fraction with a zero in the denominator does not have a defined value, hence the slope becomes **undefined**.

For instance, consider the points \((-7,2)\) and \((-7,-6)\). Using the slope formula:
  • First, identify the points: \(x_1 = -7, y_1 = 2 \) and \(x_2 = -7, y_2 = -6\).
  • Next, plug the values into the formula: \[\text{slope} = \frac{-6 - 2}{-7 - (-7)} \].
  • This simplifies to: \[\text{slope} = \frac{-8}{0} \].

Since the denominator is zero, the slope is undefined. This tells us something special about the line: it is vertical.
Undefined slopes are a clear indicator of vertical lines.
vertical line
A **vertical line** runs up and down on the graph, parallel to the y-axis.

It is a special type of line with unique properties:
  • All points on a vertical line have the same x-coordinate but different y-coordinates.
  • It does not have a defined slope (its slope is undefined).
  • The equation of a vertical line is in the form \(x = a\), where \(a\) is the x-coordinate of any point on the line.

For example, the line passing through the points \((-7,2)\) and \((-7,-6)\) is vertical because their x-coordinates are the same, which is -7.

Remember, vertical lines will always make the slope formula produce a zero in the denominator, causing the slope to be undefined. In simpler terms, vertical lines do not tilt left or right, they just go straight up and down. This constant x-value characteristic of vertical lines is incredibly useful in quickly identifying them in a set of points or an equation.

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