2.2 Linear expressions - Revision history

Eakc07 at 18:06, 31 October 2008

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

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/>

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Området y mindre än x}} +{{:2.2 - Figure - The region y less than x}})

Tekbot — Fri, 12 Sep 2008 12:34:28 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Området y mindre än x}} +{{:2.2 - Figure - The region y less than x}})

←Older revision		Revision as of 12:34, 12 September 2008
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	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/>

-	<center>{{:2.2 - ~~Figur~~ - ~~Området~~ y ~~mindre än~~ 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 dashed means that the points on the line do not belong to the coloured area.

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Området 2 ≤ 3x + 2y ≤ 4}} +{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}})

Tekbot — Fri, 12 Sep 2008 12:30:16 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Området 2 ≤ 3x + 2y ≤ 4}} +{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}})

←Older revision		Revision as of 12:30, 12 September 2008
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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 coloured areas overlap.

-	<center>{{:2.2 - ~~Figur~~ - ~~Området~~ 2 ≤ 3x + 2y ≤ 4}}</center>	+	<center>{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}}</center>
	<center><small>The figure shows the region <math>2\le 3x+2y\le 4</math>.</small></center>		<center><small>The figure shows the region <math>2\le 3x+2y\le 4</math>.</small></center>
	</div>		</div>

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Områdena 3x + 2y ≥ 2 och 3x + 2y ≤ 4}} +{{:2.2 - Figure - The regions 3x + 2y ≥ 2 and 3x + 2y ≤ 4}})

Tekbot — Fri, 12 Sep 2008 12:28:09 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Områdena 3x + 2y ≥ 2 och 3x + 2y ≤ 4}} +{{:2.2 - Figure - The regions 3x + 2y ≥ 2 and 3x + 2y ≤ 4}})

←Older revision		Revision as of 12:28, 12 September 2008
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	The points that satisfy the first inequality are on and above the line <math>y \ge 1-\tfrac{3}{2}x</math> while the points that satisfy the other inequality are on or below the line <math>y\le 2-\tfrac{3}{2}x</math>.		The points that satisfy the first inequality are on and above the line <math>y \ge 1-\tfrac{3}{2}x</math> while the points that satisfy the other inequality are on or below the line <math>y\le 2-\tfrac{3}{2}x</math>.

-	<center>{{:2.2 - ~~Figur~~ - ~~Områdena~~ 3x + 2y ≥ 2 ~~och~~ 3x + 2y ≤ 4}}</center>	+	<center>{{:2.2 - Figure - The regions 3x + 2y ≥ 2 and 3x + 2y ≤ 4}}</center>
	<center><small>The figure on the left shows the region <math>3x+2y\ge 2</math> and figure to the right shows the region <math>3x+2y\le 4</math>.</small></center>		<center><small>The figure on the left shows the region <math>3x+2y\ge 2</math> and figure to the right shows the region <math>3x+2y\le 4</math>.</small></center>

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Linjen y = kx + m}} +{{:2.2 - Figure - The line y = kx + m}})

Tekbot — Fri, 12 Sep 2008 12:27:03 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Linjen y = kx + m}} +{{:2.2 - Figure - The line y = kx + m}})

←Older revision		Revision as of 12:27, 12 September 2008
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	The graph of a linear function is always a straight line and 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 and 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.

-	<center>{{:2.2 - ~~Figur~~ - ~~Linjen~~ 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 slope ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>

Tekbot: Robot: Automated text replacement (-{{:2.2 - Figur - Linjen y = 2x - 1}} +{{:2.2 - Figure - The line y = 2x - 1}})

Tekbot — Fri, 12 Sep 2008 12:25:58 GMT

Robot: Automated text replacement (-{{:2.2 - Figur - Linjen y = 2x - 1}} +{{:2.2 - Figure - The line y = 2x - 1}})

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	{\| align="center" padding="20px"		{\| align="center" padding="20px"
-	\|align="center"\|{{:2.2 - ~~Figur~~ - ~~Linjen~~ y = 2x - 1}}	+	\|align="center"\|{{:2.2 - Figure - The line y = 2x - 1}}
	\|width="20px"\|		\|width="20px"\|
	\|align="center"\|{{:2.2 - Figure - The line y = 2 - x/2}}		\|align="center"\|{{:2.2 - Figure - The line y = 2 - x/2}}

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	{\| align="center" padding="20px"		{\| align="center" padding="20px"
-	\|align="center"\|{{:2.2 - ~~Figur~~ - ~~Linjen~~ y = 2x - 1}}	+	\|align="center"\|{{:2.2 - Figure - The line y = 2x - 1}}
	\|width="20px"\|		\|width="20px"\|
	\|align="center"\|{{:2.2 - Figure - The line y = 2 - x/2}}		\|align="center"\|{{:2.2 - Figure - The line y = 2 - x/2}}