Linear Equations - Revision history

Khanh: /* What are Linear Equations? */

2021-11-14T23:57:00Z

What are Linear Equations?

Lila: /* Licensing */

2021-10-21T17:41:06Z

Licensing

Lila: /* Parallel Lines */

2021-10-21T17:40:29Z

Parallel Lines

Lila: /* Finding It */

2021-10-21T17:39:33Z

Finding It

Lila at 17:39, 21 October 2021

2021-10-21T17:39:03Z

Lila: /* Licensing */

2021-10-21T17:37:14Z

Licensing

Lila at 17:35, 21 October 2021

2021-10-21T17:35:59Z

Lila: /* Linear Equations */

2021-10-04T16:47:16Z

Linear Equations

Lila: /* Slope */

2021-10-04T16:46:47Z

Slope

Lila: /* X and Y axis intercepts */

2021-10-04T16:46:18Z

X and Y axis intercepts

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 In the functions section we talk about how a function is like a box that takes an independent input value and uses a rule defined mathematically to create a unique output value.  The value for the output is dependent on the value that is put in the box.  We call the values that are going into the box the independent values or the domain.  We call the values coming out of the box the dependent values or the range.
-Unless we specify differently, on Cartesian graphs the domain is the real numbers.  In the  Cartesian Plane section we saw how running different values through a function to identify the points on the Cartesian plane by picking the first (x) value of the point the domain and the second (y) value from the range.  To restate this: by convention the two variables for a function on the Cartesian Plane are x for the domain, the independent variable, and y for the range, the dependent variable.  The variable y is the same as writing f(x).  Mathematicians recognize this equivalence but generally prefer to write y because its shorter.  Because a function definition has an input and an output it must also contain an equal sign.  The section in this book [[Algebra/Solving Equations|solving equations]] showed the various operations we can perform on both sides of an equal sign and still maintain the notion of equivalence.  In this section we plug different values into the independent variable and solve to find the associated dependent variable. For instance if we start with the equation:
+Unless we specify differently, on Cartesian graphs the domain is the real numbers.  In the  Cartesian Plane section we saw how running different values through a function to identify the points on the Cartesian plane by picking the first (x) value of the point the domain and the second (y) value from the range.  To restate this: by convention the two variables for a function on the Cartesian Plane are x for the domain, the independent variable, and y for the range, the dependent variable.  The variable y is the same as writing f(x).  Mathematicians recognize this equivalence but generally prefer to write y because its shorter.  Because a function definition has an input and an output it must also contain an equal sign.  The section in this book solving equations showed the various operations we can perform on both sides of an equal sign and still maintain the notion of equivalence.  In this section we plug different values into the independent variable and solve to find the associated dependent variable. For instance if we start with the equation:
 <center><math>y = x</math></center>

← Older revision		Revision as of 17:41, 21 October 2021
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	* [https://en.wikibooks.org/wiki/Algebra/Linear_Equations_and_Functions Linear Equations and Functions, Wikibooks: Algebra] under a CC BY-SA license		* [https://en.wikibooks.org/wiki/Algebra/Linear_Equations_and_Functions Linear Equations and Functions, Wikibooks: Algebra] under a CC BY-SA license
	* [https://en.wikibooks.org/wiki/Algebra/Intercepts Intercepts, Wikibooks: Algebra] under a CC BY-SA license		* [https://en.wikibooks.org/wiki/Algebra/Intercepts Intercepts, Wikibooks: Algebra] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Algebra/Slope Slope, Wikibooks: Algebra] under a CC BY-SA license

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 ⇒<math> m = 2</math>
-Therefore,
+Slope of AB = Slope of XY, so Slope of AB = 2
 ==Resources==

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 [[image:Various_Linear_Slopes.PNG]]
-== Finding It ==
 For the most part finding slope when given information is a simple matter. Simply take the slope equation y=mx+b and replace the variable with whatever information you know, and solve.

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 Horizontal lines have the form: <math>\ y = a </math> ; where a is a constant, i.e. <math> a \in R </math><br/>
 Vertical lines have the form: <math>\ x = a </math> ; where a is a constant, i.e. <math> a \in R </math>
 ==Resources==

← Older revision		Revision as of 17:37, 21 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [https://en.wikibooks.org/wiki/Algebra/Linear_Equations_and_Functions Linear Equations and Functions, Wikibooks: Algebra] under a CC BY-SA license		* [https://en.wikibooks.org/wiki/Algebra/Linear_Equations_and_Functions Linear Equations and Functions, Wikibooks: Algebra] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Algebra/Intercepts Intercepts, Wikibooks: Algebra] under a CC BY-SA license

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 Its domain is set { C<sub>1</sub> } and range is set '''R''' (unless otherwise specified). x&nbsp;=&nbsp;C<sub>1</sub> is technically not a function (there is more than one value of y for each value of x), but it's a relation. Vertical lines have '''no''' slope (m&nbsp;=&nbsp;divide by zero, undefined, plus and minus infinity). These are the only types of linear equations of the general form shown previously which are not linear functions.  The equation x&nbsp;=&nbsp;0 is '''exactly''' the y-axis. Lines with steepness approaching vertical have very large-magnitude slopes but are still functions.
-== Linear Equations ==
+== General Linear Equations ==
 We are going to start by looking at simple functions called  '''linear equations'''.  When none of the instances of x and y in the algebraic expression defining the function rule have exponents then all the instances of x and y can be combined into just two occurrences.  the graph of the expression can be represented as a straight line.  The equation that expresses the function is considered a '''linear equation''' with two variables.  The following equation is a simple example of such a linear equation:

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 When you are trying to graph a linear equation finding the axes intercepts is often the easiest way to go about doing it.  To find the x-intercept, set y = 0 and solve for x.  To find the y-intercept, set x = 0 and solve for y.  For most examples the intercepts are different points, and a line can be drawn through the two intercepts. If both intercepts are (0,0), then another point must be determined to graph the line. If the equations is in the form x = c or y = c, the horizontal or vertical lines are very simple to plot.
-===Slope===
+==Slope==
 '''Algebra/Slope'''

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 An axis intercept point is a point where the graph of a function, relation, or equation intersects the X or Y axes.  This section is about finding out how a particular set of functions: linear functions cross the axes.
-We know that the domains of most lines are infinite because they are defined at every value of X.  The exception is lines that are defined as <math>X=c</math> where c is a number we choose when we write the function.  By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function.  We've seen the graph of this relationships is a vertical line that passes through the point (c,Y).  In the picture below we show that when c=0 then our line is the same as the Y axis.  When <math> c \ne 0 </math> (as in the drawing where X = 3) then the line can never intercept the Y axis.
+We know that the domains of most lines are infinite because they are defined at every value of X.  The exception is lines that are defined as <math>X=c</math> where c is a number we choose when we write the function.  By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function.  We've seen the graph of this relationships is a vertical line that passes through the point (c,Y).
 Lines with the equation X=C intersect the X axis once and the Y axis 0 or infinite times.
-We can also restrict the range of a function by simply writing <math>Y=c</math> where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c).  When Y=o this line is the same as the X axis.  when <math>c \ne 0 </math> (as in the drawing where Y = 3)then the line can never intercept the X axis.
+We can also restrict the range of a function by simply writing <math>Y=c</math> where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c).  When Y=o this line is the same as the X axis.  when <math>c \ne 0 </math>, then the line can never intercept the X axis.
 Lines with the equation Y=C intersect the Y axis once and the X axis 0 or infinite times.
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-In this book we are going to accept the [[w:Parallel postulate|statement]] "At most one line can be drawn through any point not on a given line parallel to the given line in a plane."  There is a branch of mathematics called "non-euclidean geometry" that was founded a little more than 160 years ago.  Even if you are not interested in mathematics it is worth looking at this Wikipedia article on [[w:Geometry|''geometry'']] to get a feel for how formalizing geometry with algebraic methods and then moving beyond them has changed civilization. If you continue in a career requiring advanced mathematics such as Engineering or Physics you might want to follow your interests to see the effect of non-euclidean geometry in your career.
+In this page we are going to accept the statement"At most one line can be drawn through any point not on a given line parallel to the given line in a plane."
 We've seen that for the equation Y=mX + b the Y intercept will always be at b because that is where X=0.