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Chapter 0: Problem 96

True or False? In Exercises \(93-96,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The equation of any line can be written in general form.

Short Answer

Expert verified
The statement is true, any line's equation can be written in general form.

Step by step solution

01

Identify the General Form

Firstly, we need to identify what is meant by 'general form'. The 'General Form' of the equation of a line is usually written as: \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants.
02

Analyze Statement

The second step involves analyzing the statement. In theory, it is true that you can write the equation of a line in a general form, assuming you allow \(A\), \(B\), and \(C\) to be any real number, including zero, unless specified otherwise.
03

Consider Line Restrictions

In addition, a line in the two-dimensional plane is typically represented by a linear equation. Therefore, unless there are some unusual restrictions on the coefficients being used, virtually every line can be represented in this form.
04

Formulate Conclusion

Based on the above understanding, the statement is true.

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Key Concepts

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

General Form
In mathematics, the general form of the equation of a line is one of the most common ways to express a linear equation. The general form is written as \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables.
This form is very flexible because it can describe any straight line in a two-dimensional plane, if you choose appropriate values for \(A\), \(B\), and \(C\).

Here are some key points about the general form:
  • It is a standardized way to write linear equations, making it easier to handle multiple lines at once.
  • The constants \(A\), \(B\), and \(C\) can be any real numbers. Importantly, \(A\) and \(B\) cannot both be zero, as this would not represent a line.
  • For horizontal lines, \(B = 0\); for vertical lines, \(A = 0\).
Thus, the general form provides a comprehensive way to express linearity across different equations.
Equation of a Line
The equation of a line is fundamental in geometry and algebra, describing a straight line on a graph. There are several ways to express an equation of a line, including slope-intercept form (\(y = mx + b\)), point-slope form, and others.
The general form (\(Ax + By + C = 0\)) is one such method, offering a standard way to articulate these lines.

Some advantages of using the general form are:
  • It neatly accommodates lines that have vertical or horizontal orientations, something slope-intercept form isn't well-suited for.
  • When rearranged, it can help quickly identify intercepts and important line characteristics.
  • Having the equation in general form is useful for computational solutions and is easily understandable in computer algorithms.
This versatile form of line equation covers all potential cases, making it invaluable in mathematical analyses.
Two-Dimensional Plane
A two-dimensional plane is an essential concept in geometry. It is a flat surface extending infinitely in two dimensions, typically defined within a Cartesian coordinate system with an \(x\) and \(y\) axis.
Every point on the plane is defined by an ordered pair \((x, y)\), which combines to systematically describe locations on this plane.

Key things to know about two-dimensional planes:
  • Lines, which are described by linear equations, are sets of points that extend infinitely in both directions on the plane.
  • The intersection of two lines on a plane can provide valuable information, such as the solution to a system of equations.
  • The simplicity of a 2D plane makes it a foundational tool for introducing concepts of linear algebra and calculus.
Understanding how lines and points interact within a two-dimensional plane is essential for exploring more advanced mathematical topics.

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Most popular questions from this chapter

Straight-Line Depreciation A small business purchases a piece of equipment for \(\$ 875\) . After 5 years, the equipment will be outdated, having no value. (a) Write a linear equation giving the value \(y\) of the equipment in terms of the time \(x\) (in years), \(0 \leq x \leq 5 .\) (b) Find the value of the equipment when \(x=2\) (c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is \(\$ 200\) .

Modeling Data An instructor gives regular 20 -point quizzes and 100 -point exams in a mathematics course. Average scores for six students, given as ordered pairs \((x, y)\) where \(x\) is the average quiz score and \(y\) is the average exam score, are \((18,87),(10,55),(19,96),(16,79),(13,76),\) and \((15,82) .\) (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use a graphing utility to plot the points and graph the regression line in the same viewing window. (c) Use the regression line to predict the average exam score for a student with an average quiz score of \(17 .\)

Distance In Exercises \(83-86\) , find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+\) \(C=0 .\) $$ =\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}} $$ Line: \(3 x-4 y=1\) Line: \(3 x-4 y=10\)

Sketching a Graph of a Function In Exercises \(33-40\) , sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ g(t)=3 \sin \pi t $$

Proof Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular.
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