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Chapter 1: Problem 5

Find the slope of the line that passes through the given points. $$(0,0) \text { and }(0,4)$$

Short Answer

Expert verified
The slope of the line passing through the points (0,0) and (0,4) is undefined.

Step by step solution

01

Assign Points

First, assign the points to the respective variables: x1=0, y1=0, x2=0, y2=4.
02

Apply Slope Formula

Next, apply the slope formula \( m=\frac{y_2-y_1}{x_2-x_1} \). Substituting the assigned values, we have \( m=\frac{4-0}{0-0} \).
03

Compute Result

Simplify the terms in the formula, it results in a division by zero which is not defined, thus the slope is undefined.

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Key Concepts

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

Point-Slope Formula
The Point-Slope Formula is an essential tool in geometry and algebra. It's used to find the slope of a line when given two points on that line. The formula is represented as \( m=\frac{y_2-y_1}{x_2-x_1} \), where \( m \) is the slope, \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points.

This formula calculates how steep a line is by comparing the vertical change (change in \( y \) values) to the horizontal change (change in \( x \) values) between these two points.

Key points to remember about this formula include:
  • Subtract the first point's \( y \) value from the second to find the vertical change.
  • Subtract the first point's \( x \) value from the second to calculate the horizontal change.
  • Divide the vertical change by the horizontal change to get the slope.
It is crucial to perform these steps correctly to avoid mistakes, especially when dealing with fractions.
Division by Zero
Division by zero is a concept in mathematics that occurs when a number is divided by zero. In the context of the slope of a line, when you have a zero in the denominator as seen in the Slope Formula, it indicates a potential mathematical problem.

Mathematically, dividing by zero is undefined because it doesn't produce a number or result that fits within our number system. A primary reason is that zero times any number is always zero, making it impossible to determine a meaningful single number that represents the division.

A practical understanding includes:
  • Whenever you encounter zero in the denominator part of a fraction, it signals division by zero.
  • This is crucial when working with the slope of a line, as an undefined result here suggests particular properties of the line itself.
  • Understanding division by zero helps prevent errors in computation and interpreting mathematical results accurately.
This concept plays a significant role in topics where division is a common operation, such as algebra and calculus.
Undefined Slope
An undefined slope is a special case that occurs when trying to determine the slope of a vertical line. In our original exercise, calculating the slope resulted in division by zero.

This situation provides important insights into the characteristics of the line in question. Specifically, a vertical line does not have a defined steepness or slope in the conventional sense because it goes straight up and down.

Key characteristics of vertical lines or lines with undefined slopes include:
  • All points on the line have the same \( x \) value, indicating no horizontal change.
  • The slope formula produces a division by zero, thus the slope is categorized as undefined.
  • This concept illustrates how vertical lines behave differently, lacking a conventional slope measure.
Understanding undefined slopes is crucial for correctly interpreting the behavior and structure of vertical lines on a graph.

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