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Chapter 7: Problem 71

What does it mean if the slope of a line is undefined?

Short Answer

Expert verified
The slope of a line is undefined when the line is vertical, meaning there's no horizontal change. In terms of calculation, it becomes undefined because we are trying to divide by zero, which is undefined in mathematics.

Step by step solution

01

Understanding Slope

The slope of a line is a measure of how steep the line is. It is calculated as the vertical change (the 'rise') divided by the horizontal change (the 'run'). The slope is usually represented as 'm' in the equation of a straight line, which is 'y = mx + c', where 'c' is the y-intercept.
02

Undefined Slope

A slope is undefined when there is no horizontal change, meaning the 'run' is zero. In mathematics, division by zero is undefined.
03

Visual Aspect of Undefined Slope

Visually, a line with an undefined slope is a vertical line. No matter how much we move up or down along the line, we won't move left or right, hence the horizontal change (run) is zero. Since the slope 'm' is calculated as 'rise/run', and here the run is zero, the slope becomes undefined.

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Key Concepts

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

undefined slope
In mathematics, the slope of a line quantifies how steep the line is. When the slope is "undefined," it doesn't mean that we don't understand or can't know the slope. It actually has a very specific meaning.

An undefined slope occurs when we attempt to divide by zero during our slope calculations. If we think of the slope formula, which is the "rise over run," the "run" in the denominator represents the horizontal change between any two points on the line. If a line makes no horizontal movement鈥攎eaning it is perfectly vertical鈥攖he "run" equals zero.
  • A zero in the denominator of the slope formula \(m = \frac{\text{rise}}{\text{run}}\) results in an undefined slope.
  • Remember, division by zero in mathematics is not allowed, making our slope undefined.
  • In this case, the slope does not take on any finite value, as it's not possible to divide by zero. Therefore, we describe the slope of such lines as simply "undefined."
vertical line
A vertical line is one of the simplest types of lines to visualize.

Imagine a line that runs straight up and down, just like the sides of a tall building. It does not tilt to the left or right at all. In mathematical terms, this means:
  • The horizontal change鈥攐r "run"鈥攊s zero.
  • Because of this lack of horizontal change, the line always has an undefined slope.
  • Every point on the line shares the same x-coordinate. For example, if every point on the line has an x-coordinate of 5, you can describe the line with the equation \(x = 5\).
This characteristic of spanning only vertically without any horizontal shift clearly establishes why the slope of a vertical line ends up being undefined.
equation of a straight line
The equation of a straight line is crucial in geometry and provides a clear mathematical representation of a line. It typically appears in the form \(y = mx + c\), where:
  • \(m\) represents the slope of the line. It indicates how steep the line is, with positive values indicating an upward tilt from left to right, and negative values a downward tilt.
  • \(c\) is the y-intercept, indicating where the line crosses the y-axis.
However, for vertical lines, this formula doesn't apply directly since the slope \(m\) is undefined. Instead:
  • The equation simplifies to \(x = a\), where \(a\) is the fixed x-coordinate for all points on the line.
  • This equation states the consistent x-coordinate, making no reference to \(y\).
It's essential to recognize this unique case for vertical lines when working with the equations of straight lines.
division by zero
In mathematics, division by zero is a forbidden operation because it defies the fundamental rules of arithmetic. If you imagine dividing a pie amongst people, dividing by zero asks you to distribute it to nobody, which makes no sense.

This concept comes into play specifically when calculating the slope of a vertical line:
  • The slope formula is \(m = \frac{\text{rise}}{\text{run}}\).
  • For vertical lines, the "run" or horizontal change is zero, leading to a division by zero scenario.
Since division by zero is undefined in mathematics, whenever it happens in calculations like when determining the slope of a vertical line, we describe the result as "undefined."

Understanding this helps avoid practical errors in mathematical computations and reinforces the concept that certain operations just don't produce valid numerical results.

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Most popular questions from this chapter

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Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}y<-2 x+4 \\ y

On June 24,1948 , the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was 30,000 cubic feet for an American plane and 20,000 cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity, but were subject to the following restrictions: \- No more than 44 planes could be used. \- The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \- The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000\). Total weekly costs could not exceed \(\$ 300,000\). Find the number of American and British planes that were used to maximize cargo capacity.

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