2.2 Linear expressions - Revision history

Phil: \cdot to \times

Phil — Tue, 06 Jan 2009 15:11:50 GMT

\cdot to \times

←Older revision		Revision as of 15:11, 6 January 2009
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	This gives the base of the triangle as <math>4</math> units of length and a height of <math>2</math> units of length.		This gives the base of the triangle as <math>4</math> units of length and a height of <math>2</math> units of length.

-	The area of this triangle is therefore <math> 4\~~cdot~~ 2/2=4</math> square units.	+	The area of this triangle is therefore <math> 4\times 2/2=4</math> square units.
	</div>		</div>

Phil: kx+m becomes mx+c throughout

Phil — Tue, 06 Jan 2009 15:10:22 GMT

kx+m becomes mx+c throughout

←Older revision		Revision as of 15:10, 6 January 2009
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	*Solve equations that are linear, or linear when simplified.		*Solve equations that are linear, or linear when simplified.
-	*Convert between the forms ''y'' = ''kx'' + ''m'' and ''ax'' + ''by'' + ''c'' = 0.	+	*Convert between the forms ''y'' = ''mx'' + ''c'' and ''ax'' + ''by'' + ''c'' = 0.
	*Sketch straight lines from their equations.		*Sketch straight lines from their equations.
	* Solve geometric problems that contain straight lines.		* Solve geometric problems that contain straight lines.
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	<div class="regel">		<div class="regel">
-	{{Displayed math\|\|<math>y = kx+m</math>}}	+	{{Displayed math\|\|<math>y = mx+c</math>}}
	</div>		</div>

-	where <math>k</math> and <math>m</math> are constants.	+	where <math>m</math> and <math>c</math> are constants.

-	The graph of a linear function is always a straight line. The constant <math>k</math> indicates the gradient of the line with respect to the <math>x</math>-axis and <math>m</math> gives the intercept: that is, the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.	+	The graph of a linear function is always a straight line. The constant <math>m</math> indicates the gradient of the line with respect to the <math>x</math>-axis and <math>c</math> gives the intercept: that is, the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.

	<center>{{:2.2 - Figure - The line y = mx + c}}</center>		<center>{{:2.2 - Figure - The line y = mx + c}}</center>
	<center><small>The line ''y'' = ''mx'' + ''c'' has gradient ''m'' and cuts the ''y''-axis at (0,''c'')</small></center>		<center><small>The line ''y'' = ''mx'' + ''c'' has gradient ''m'' and cuts the ''y''-axis at (0,''c'')</small></center>

-	The constant <math>k</math> is called the gradient. To state that a straight line has gradient <math>k</math> means that for each increase of one unit in the positive <math>x</math>-direction there is an increase of <math>k</math> units in the positive <math>y</math>-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that <math>y</math> decreases as <math>x</math> increases: that is, if	+	The constant <math>m</math> is called the gradient. To state that a straight line has gradient <math>m</math> means that for each increase of one unit in the positive <math>x</math>-direction there is an increase of <math>m</math> units in the positive <math>y</math>-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that <math>y</math> decreases as <math>x</math> increases: that is, if

-	*<math>k>0\,</math> the line slopes upwards,	+	*<math>m>0\,</math> the line slopes upwards,
-	*<math>k<0\,</math> the line slopes downwards.	+	*<math>m<0\,</math> the line slopes downwards.

-	For a horizontal line (parallel to the <math>x</math>-axis) <math>k=0</math>, whereas a vertical line (parallel to the <math>y</math>-axis) does not have a <math>k</math> value ( a vertical line cannot be written in the form <math>y=kx+m</math>).	+	For a horizontal line (parallel to the <math>x</math>-axis) <math>m=0</math>, whereas a vertical line (parallel to the <math>y</math>-axis) does not have a <math>m</math> value ( a vertical line cannot be written in the form <math>y=mx+c</math>).

	<div class="exempel">		<div class="exempel">
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	<ol type="a">		<ol type="a">
	<li> Sketch the line <math>y=2x-1</math>. <br/><br/>		<li> Sketch the line <math>y=2x-1</math>. <br/><br/>
-	Comparing with the standard equation <math>y=kx+m</math>, we see that <math>k=2</math> and <math>m=-1</math>. This tells us that the line's gradient is <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below-left.</li>	+	Comparing with the standard equation <math>y=mx+c</math>, we see that <math>m=2</math> and <math>c=-1</math>. This tells us that the line's gradient is <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below-left.</li>
	<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>		<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>
-	The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math>. From this we see that has the gradient <math>k= -\tfrac{1}{2}</math>and that the intercept <math>m=2</math>. See the figure below to the right.	+	The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math>. From this we see that has the gradient <math>m= -\tfrac{1}{2}</math>and that the intercept <math>c=2</math>. See the figure below to the right.
	</ol>		</ol>

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-	If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction correspond to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>k=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's gradient is <math>k= \frac{2}{3}</math>.	+	If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction correspond to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>m=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's gradient is <math>m= \frac{2}{3}</math>.

	<center>python</center>		<center>python</center>
	</div>		</div>

-	Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>~~k_1~~</math> and <math>~~k_2~~</math> are perpendicular to one another, then <math>~~k_2~~ = -\frac{1}{~~k_1~~}</math>, which also can be written as <math>~~k_1 k_2~~ = -1</math>.	+	Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>m_1</math> and <math>m_2</math> are perpendicular to one another, then <math>m_2 = -\frac{1}{m_1}</math>, which also can be written as <math>m_1 m_2 = -1</math>.

	<center>{{:2.2 - Figure - The gradients of perpendicular lines}}</center>		<center>{{:2.2 - Figure - The gradients of perpendicular lines}}</center>

-	The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction	+	The straight line in the figure on the left has gradient <math>m</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>m</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{m}</math> because now <math>-m</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction

	<div class="exempel">		<div class="exempel">
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	<li>Put the line <math>y=5x+7</math> into the form <math>ax+by=c</math>.<br/><br/>		<li>Put the line <math>y=5x+7</math> into the form <math>ax+by=c</math>.<br/><br/>
	Move the <math>x</math>-term to the left-hand side: <math>-5x+y=7</math>.</li>		Move the <math>x</math>-term to the left-hand side: <math>-5x+y=7</math>.</li>
-	<li> Put the line <math>2x+3y=-1</math> into the form <math>y=kx+m</math>.<br/><br/>	+	<li> Put the line <math>2x+3y=-1</math> into the form <math>y=mx+c</math>.<br/><br/>
	Move the <math>x</math>-term to the right-hand side <math>3y=-2x-1 </math> and divide both sides by <math>3</math>		Move the <math>x</math>-term to the right-hand side <math>3y=-2x-1 </math> and divide both sides by <math>3</math>
	{{Displayed math\|\|<math>y=-\frac{2}{3}x - \frac{1}{3}\,\mbox{.}</math>}}		{{Displayed math\|\|<math>y=-\frac{2}{3}x - \frac{1}{3}\,\mbox{.}</math>}}
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	[http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml '''Here'''] you can see how an equation for a line can be obtained if we know the coordinates of two points on the line.		[http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml '''Here'''] you can see how an equation for a line can be obtained if we know the coordinates of two points on the line.

-	[http://www.theducation.se/hemsida//gymnasium_komvux/webbaserade_laromedel_och_webbstod/matematik_3000/experimentera_med_den_rata_linjen/index.asp '''Here'''] you can vary k and m and see how this affects the line's characteristics.	+	[http://www.theducation.se/hemsida//gymnasium_komvux/webbaserade_laromedel_och_webbstod/matematik_3000/experimentera_med_den_rata_linjen/index.asp '''Here'''] you can vary gradient and intercept and see how this affects the line's characteristics.

Tek at 14:48, 6 January 2009

Tek — Tue, 06 Jan 2009 14:48:09 GMT

←Older revision		Revision as of 14:48, 6 January 2009
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	Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.		Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.

-	<center>{{:2.2 - Figure - The ~~slope~~ of perpendicular lines}}</center>	+	<center>{{:2.2 - Figure - The gradients of perpendicular lines}}</center>

	The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction		The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction

Tek at 12:10, 6 January 2009

Tek — Tue, 06 Jan 2009 12:10:25 GMT

←Older revision		Revision as of 12:10, 6 January 2009
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	The graph of a linear function is always a straight line. The constant <math>k</math> indicates the gradient of the line with respect to the <math>x</math>-axis and <math>m</math> gives the intercept: that is, the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.		The graph of a linear function is always a straight line. The constant <math>k</math> indicates the gradient of the line with respect to the <math>x</math>-axis and <math>m</math> gives the intercept: that is, the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.

-	<center>{{:2.2 - Figure - The line y = kx + m}}</center>	+	<center>{{:2.2 - Figure - The line y = mx + c}}</center>
-	<center><small>The line ''y'' = ''kx'' + ''m'' has gradient ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>	+	<center><small>The line ''y'' = ''mx'' + ''c'' has gradient ''m'' and cuts the ''y''-axis at (0,''c'')</small></center>

	The constant <math>k</math> is called the gradient. To state that a straight line has gradient <math>k</math> means that for each increase of one unit in the positive <math>x</math>-direction there is an increase of <math>k</math> units in the positive <math>y</math>-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that <math>y</math> decreases as <math>x</math> increases: that is, if		The constant <math>k</math> is called the gradient. To state that a straight line has gradient <math>k</math> means that for each increase of one unit in the positive <math>x</math>-direction there is an increase of <math>k</math> units in the positive <math>y</math>-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that <math>y</math> decreases as <math>x</math> increases: that is, if
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	Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.		Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.

-	<center>{{:2.2 - Figure - The ~~gradients~~ of perpendicular lines}}</center>	+	<center>{{:2.2 - Figure - The slope of perpendicular lines}}</center>

	The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction		The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction

Phil: Edited for local idiom, notation and terminology; note in partic. the use of "gradient" instead of the American "slope". Problem with kx+m vs mx+c remains to be solved; advice awaited.

Phil — Wed, 31 Dec 2008 18:50:54 GMT

Edited for local idiom, notation and terminology; note in partic. the use of "gradient" instead of the American "slope". Problem with kx+m vs mx+c remains to be solved; advice awaited.

←Older revision		Revision as of 18:50, 31 December 2008
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	{{Info\|		{{Info\|
	'''Contents:'''		'''Contents:'''
-	*~~First degree~~ equations	+	*Linear equations
	* Equation of a straight line		* Equation of a straight line
	*Geometrical problems		*Geometrical problems
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	After this section you will have learned how to:		After this section you will have learned how to:

-	*Solve ~~algebraic~~ equations, ~~which after simplification results in first degree equations~~.	+	*Solve equations that are linear, or linear when simplified.
	*Convert between the forms ''y'' = ''kx'' + ''m'' and ''ax'' + ''by'' + ''c'' = 0.		*Convert between the forms ''y'' = ''kx'' + ''m'' and ''ax'' + ''by'' + ''c'' = 0.
-	*Sketch straight lines from their ~~equation~~.	+	*Sketch straight lines from their equations.
	* Solve geometric problems that contain straight lines.		* Solve geometric problems that contain straight lines.
	*Sketch regions defined by linear inequalities and determine the area of these regions.		*Sketch regions defined by linear inequalities and determine the area of these regions.
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	== First degree equations ==		== First degree equations ==

-	To solve ~~first degree~~ equations (also known as ~~linear~~ equations) we perform ~~calculations~~ on both sides simultaneously~~. This~~ gradually ~~simplifies~~ the equation and ultimately ~~leads~~ to <math>x</math> being alone on one side of ~~the equation~~.	+	To solve linear equations (also known as equations of degree one) we perform operations on both sides simultaneously, in such a way as to gradually simplify the equation and ultimately lead to <math>x</math> being alone on one side of it.

	<div class="exempel">		<div class="exempel">
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	Divide both sides by <math>3</math>		Divide both sides by <math>3</math>
	:<math>\frac{3x}{3} = \frac{6}{3}\,</math>.		:<math>\frac{3x}{3} = \frac{6}{3}\,</math>.
-	After ~~having cancelled~~ <math>3</math> on the left-hand side we have	+	After cancelling <math>3</math> on the left-hand side we have
	:<math> x=\frac{6}{3} = 2</math>.</li>		:<math> x=\frac{6}{3} = 2</math>.</li>

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	</div>		</div>

-	A ~~first degree~~ equation can be written in the ~~normal~~ form <math>ax=b</math>. The solution is then simply <math>x=b/a</math> (we must assume that <math>a\not=0</math>).	+	A linear equation can always be written in the form <math>ax=b</math>. The solution is then simply <math>x=b/a</math> (we must assume that <math>a\not=0</math>).
-	~~The possible difficulties that may occur when you solve a first degree equation are thus not in themselves the form of the solution~~, ~~but rather the simplifications that may be~~ needed to ~~achieve~~ the ~~normal~~ form. ~~Below~~ are a couple examples ~~in which the equation can be simplified to a linear normal form, thus having a unique solution~~.	+	Sometimes, however, some work is needed in order to place the equation in this form. Here are a couple of examples.
	<div class="exempel">		<div class="exempel">
	'''Example 2'''		'''Example 2'''
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	</div>		</div>

-	It is not always obvious that we are dealing with a ~~first degree~~ equation. In the following two examples simplifications ~~turn~~ the original equation into a ~~first degree equation~~.	+	It is not always obvious that we are dealing with a linear equation. In the following two examples simplifications are needed to convert the original equation into a linear one.

	<div class="exempel">		<div class="exempel">
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	where <math>k</math> and <math>m</math> are constants.		where <math>k</math> and <math>m</math> are constants.

-	The graph of a linear function is always a straight line. The constant <math>k</math> indicates the ~~slope~~ of the line with respect to the <math>x</math>-axis and <math>m</math> gives the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.	+	The graph of a linear function is always a straight line. The constant <math>k</math> indicates the gradient of the line with respect to the <math>x</math>-axis and <math>m</math> gives the intercept: that is, the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.

	<center>{{:2.2 - Figure - The line y = kx + m}}</center>		<center>{{:2.2 - Figure - The line y = kx + m}}</center>
-	<center><small>The line ''y'' = ''kx'' + ''m'' has ~~slope~~ ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>	+	<center><small>The line ''y'' = ''kx'' + ''m'' has gradient ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>

-	The constant <math>k</math> is called the ~~slope and states~~ that ~~given~~ a unit ~~change~~ in the positive <math>x</math>-direction~~, along the line~~ there is a <math>k</math> ~~unit change~~ in the positive <math>y</math>-direction. ~~Thus~~ if	+	The constant <math>k</math> is called the gradient. To state that a straight line has gradient <math>k</math> means that for each increase of one unit in the positive <math>x</math>-direction there is an increase of <math>k</math> units in the positive <math>y</math>-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that <math>y</math> decreases as <math>x</math> increases: that is, if

	*<math>k>0\,</math> the line slopes upwards,		*<math>k>0\,</math> the line slopes upwards,
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	<ol type="a">		<ol type="a">
	<li> Sketch the line <math>y=2x-1</math>. <br/><br/>		<li> Sketch the line <math>y=2x-1</math>. <br/><br/>
-	Comparing with the standard equation <math>y=kx+m</math>, we see that <math>k=2</math> and <math>m=-1</math>. This ~~gives~~ the line's ~~slope as~~ <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below-left.</li>	+	Comparing with the standard equation <math>y=kx+m</math>, we see that <math>k=2</math> and <math>m=-1</math>. This tells us that the line's gradient is <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below-left.</li>
	<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>		<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>
-	The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math>. From this we see that has the ~~slope~~ <math>k= -\tfrac{1}{2}</math>and that <math>m=2</math>. See the figure below to the right.	+	The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math>. From this we see that has the gradient <math>k= -\tfrac{1}{2}</math>and that the intercept <math>m=2</math>. See the figure below to the right.
	</ol>		</ol>

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	''' Example 7'''		''' Example 7'''

-	What is the ~~slope~~ of the straight line that passes through the points <math>(2,1)</math> and <math>(5,3)</math>?	+	What is the gradient of the straight line that passes through the points <math>(2,1)</math> and <math>(5,3)</math>?


-	If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction ~~corresponds~~ to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>k=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's ~~slope~~ is <math>k= \frac{2}{3}</math>.	+	If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction correspond to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>k=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's gradient is <math>k= \frac{2}{3}</math>.

	<center>python</center>		<center>python</center>
	</div>		</div>

-	Two straight lines that are parallel clearly have the same ~~slope~~. It is also possible to see (such as in the figure below) that ~~for~~ two lines with ~~slopes~~ <math>k_1</math> and <math>k_2</math> ~~that~~ are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.	+	Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.

-	<center>{{:2.2 - Figure - The ~~slope~~ of perpendicular lines}}</center>	+	<center>{{:2.2 - Figure - The gradients of perpendicular lines}}</center>

-	The straight line in the figure on the left has ~~slope~~ <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has ~~slope~~ <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction	+	The straight line in the figure on the left has gradient <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has gradient <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction

	<div class="exempel">		<div class="exempel">
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	<ol type="a">		<ol type="a">
	<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>		<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>
-	The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal or greater than <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>	+	The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal to, or greater than, <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>
	<center>{{:2.2 - Figure - The region y ≥ 2}}</center></li>		<center>{{:2.2 - Figure - The region y ≥ 2}}</center></li>
	<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>		<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>
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	<center>{{:2.2 - Figure - The region y less than x}}</center>		<center>{{:2.2 - Figure - The region y less than x}}</center>

-	The fact that the line <math>y=x</math> is dashed ~~means~~ that the points on the line do not belong to the ~~coloured~~ area.	+	The fact that the line <math>y=x</math> is shown dashed signifies that the points on the line do not belong to the shaded area.
	</ol>		</ol>
	</div>		</div>
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-	Points that satisfy both inequalities form a band-like region where both ~~coloured~~ areas overlap.	+	Points that satisfy both inequalities form a band-like region where both shaded areas overlap.

	<center>{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}}</center>		<center>{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}}</center>
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	''' Example 12'''		''' Example 12'''

-	If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in ~~a coordinate system~~.	+	If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in the <math>xy</math> plane.

	<center>{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}}</center>		<center>{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}}</center>

Eakc07 at 18:06, 31 October 2008

Eakc07 — Fri, 31 Oct 2008 18:06:51 GMT

(Difference between revisions)

Jdinglis at 13:59, 26 October 2008

Jdinglis — Sun, 26 Oct 2008 13:59:42 GMT

←Older revision		Revision as of 13:59, 26 October 2008
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-	'''Keep in mind that ... '''	+	'''Keep in mind that... '''

-	~~Draw~~ your own diagrams when you solve geometrical problems and draw carefully and accurately! A good diagram can mean you are halfway to a solution, but a poor diagram may well fool you.	+	You should draw your own diagrams when you solve geometrical problems and draw carefully and accurately! A good diagram can mean you are halfway to a solution, but a poor diagram may well fool you.

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Riktningskoefficient för vinkelräta linjer}} +{{:2.2 - Figure - The slope of perpendicular lines}})

Tekbot — Mon, 15 Sep 2008 13:01:41 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Riktningskoefficient för vinkelräta linjer}} +{{:2.2 - Figure - The slope of perpendicular lines}})

←Older revision		Revision as of 13:01, 15 September 2008
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	Two straight lines that are parallel clearly have the same slope. It is also possible to see (such as in the figure below) that for two lines having slopes <math>k_1</math> and <math>k_2</math> and also are perpendicular then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.		Two straight lines that are parallel clearly have the same slope. It is also possible to see (such as in the figure below) that for two lines having slopes <math>k_1</math> and <math>k_2</math> and also are perpendicular then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.

-	<center>{{:2.2 - ~~Figur~~ - ~~Riktningskoefficient för vinkelräta linjer~~}}</center>	+	<center>{{:2.2 - Figure - The slope of perpendicular lines}}</center>

	The straight line in the figure on the left has slope <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise, we get the line in the figure to the right, and that line has slope <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction		The straight line in the figure on the left has slope <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise, we get the line in the figure to the right, and that line has slope <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Triangel begränsad av y = x, y = 2 och y = -x}} +{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}})

Tekbot — Fri, 12 Sep 2008 12:37:43 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Triangel begränsad av y = x, y = 2 och y = -x}} +{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}})

←Older revision		Revision as of 12:37, 12 September 2008
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	If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in a coordinate system.		If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in a coordinate system.

-	<center>{{:2.2 - ~~Figur~~ - ~~Triangel begränsad av~~ y = x, y = 2 ~~och~~ y = -x}}</center>	+	<center>{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}}</center>

	We find that for a point to lie in this triangle, it has to satisfy certain conditions.		We find that for a point to lie in this triangle, it has to satisfy certain conditions.

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Området y ≥ 2}} +{{:2.2 - Figure - The region y ≥ 2}})

Tekbot — Fri, 12 Sep 2008 12:35:33 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Området y ≥ 2}} +{{:2.2 - Figure - The region y ≥ 2}})

←Older revision		Revision as of 12:35, 12 September 2008
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	<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>		<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>
	The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal or greater than <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>		The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal or greater than <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>
-	<center>{{:2.2 - ~~Figur~~ - ~~Området~~ y ≥ 2}}</center></li>	+	<center>{{:2.2 - Figure - The region y ≥ 2}}</center></li>
	<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>		<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>
	A point <math>(x,y)</math> that satisfies the inequality <math>y < x</math> must have an <math>x</math>-coordinate that is larger than its <math>y</math>-coordinate. Thus the area consists of all the points to the right of the line <math>y=x</math>.<br/>		A point <math>(x,y)</math> that satisfies the inequality <math>y < x</math> must have an <math>x</math>-coordinate that is larger than its <math>y</math>-coordinate. Thus the area consists of all the points to the right of the line <math>y=x</math>.<br/>