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Chapter 1: Problem 80

True or False: Every line has a slope.

Short Answer

Expert verified
False.

Step by step solution

01

Understanding What a Line Is

A line is a straight one-dimensional figure having no thickness and extends infinitely in both directions. It can easily be represented by the equation of a line, typically written in the form, for example, \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
02

Defining Slope

The slope of a line (\(m\)) is a measure of how steep the line is. It is calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line, \( \frac{y_2 - y_1}{x_2 - x_1} \), assuming \(x_1 eq x_2\).
03

Exception: Vertical Line

In the case of a vertical line, the equation is of the form \(x = c\), where \(c\) is a constant. For a vertical line, the change in \(x\) is zero (since \(x_2 - x_1 = 0\)) and the slope is undefined, as it involves division by zero.
04

Conclusion: Lines with Undefined Slope

Since lines can be vertical and have an undefined slope, not every line has a defined slope. Therefore, the statement that every line has a slope is false.

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Key Concepts

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

Equation of a Line
An equation of a line is a mathematical expression that describes a straight line. It's commonly written in the format of \(y = mx + b\). Here, \(m\) is the slope of the line, representing how steep or slanted the line is. The \(b\) is the y-intercept, indicating where the line crosses the y-axis.
When working with the equation of a line:
  • The slope \(m\) tells us the direction and angle of the line. A positive slope means the line moves upward as it goes right, while a negative slope shows it moves downward.
  • The y-intercept \(b\) helps in plotting the initial point of the line on the graph where it meets the y-axis.
Understanding these components is essential for graphing lines and analyzing their behavior on a coordinate plane.
Undefined Slope
The term "undefined slope" comes into play when dealing with a specific type of line. In general concepts, the slope \(m\) is defined as \( \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \). However, what happens when \(x_1 = x_2\)?When this occurs, we are dealing with a vertical line. For such lines:
  • The "run" or horizontal change between two points is zero, i.e., \(x_2 - x_1 = 0\).
  • Mathematically, division by zero is undefined, so the slope becomes undefined.
  • This is why vertical lines on a slope formula perspective have an undefined slope.
Therefore, not all lines have a defined slope, and this leads to understanding why the statement "every line has a slope" is false.
Vertical Line
A vertical line is an exception in linear equations. It is represented by an equation of the form \(x = c\), where \(c\) is a constant.Key characteristics of vertical lines include:
  • They run "up and down" on the graph.
  • All points on a vertical line have the same \(x\)-coordinate, which is \(c\). This lack of horizontal change results in a vertical orientation.
  • These lines do not intersect the y-axis, unless at infinity, which is not typically represented in basic graph plotting.
This decorum illuminates why vertical lines have an undefined slope and modifies our understanding of lines that break the typical slope equation scenarios.

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Most popular questions from this chapter

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ f(x)=2 x^{2}-5 x+1 $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ \begin{array}{l} f(x)=x^{3} \\ \text { [Hint: Use } \left.(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right] \end{array} $$

Graph each function. $$ f(x)=\left\\{\begin{array}{ll} 2 x-7 & \text { if } x \geq 4 \\ 2-x & \text { if } x<4 \end{array}\right. $$

The use of cellular telephones has been growing rapidly, in both developed and developing countries. The number of cell phones worldwide (in millions) is approximated by the exponential function \(f(x)=650(1.45)^{x}\), where \(x\) is the number of years since \(2000 .\) Predict the number of cell phones in the year 2010 .
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