## Main.Systems2 History

September 27, 2008, at 04:42 AM by LFS -
Changed lines 1-2 from:
(:title Systems of Linear Equations 2х2 - 2 equations in 2 variables:)
to:
(:title Good aschiiSVG graphs Systems of Linear Equations 2х2 - 2 equations in 2 variables:)
September 27, 2008, at 03:49 AM by LFS -
(:title Systems of Linear Equations 2х2 - 2 equations in 2 variables:)

>>nav_bar<<%reg%Mathcasts&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;%exa%Examples&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;%pra%Practice%%&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[[ProblemsT/WindCurrent|%mor%More]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;%up%[[glossary/S]]
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>>&<<[[<<]]

>>-<<
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%def b%Definition%%: Given 2 linear equations in 2 unknowns (variables). Finding a solution - that is, values for both variables that make both equations true - is called solving a %red b%2х2 system of linear equations%%.
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>>-<<
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[]
(:showhide init=show div=div1 lshow="+" lhide="-":) Example of a linear system of 2 equations in 2 unknowns.
(:div1 id=div1 :)
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[]
Sample 2x2 linear system: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right.$}
>>&<<The solution to this system is {$(x,y)=(1,2)$}.
>>&<<To see this we substitute {$\color{purple}{x=1}$} and {$\color{teal}{y=2}$} and check whether both equations are "true":
-> >>&<< {$\begin{array}{c} \color{purple}{1}\color{blue}{+}\color{teal}{2}\color{blue}{ \cdot 2 \,≟\, 5} \\ \color{blue}{1 + 4 \,≟\, 5} \\ \color{blue}{5 \,=\, 5} \\ \end{array}$} {$\begin{array}{c} \color{red}{4\cdot} \color{purple}{1} \color{red}{- 2\cdot} \color{teal}{2}\color{red}{ \,≟\, 0} \\ \color{red}{4 - 4 \,≟\, 0} \\ \color{red}{0 = 0} \\ \end{array}$}
[]Graphically, we have:
{gg width=240; height=180; xmin=-3; xmax=5; xscl=1; ymin=-1; ymax=5; yscl=1; axes(); fontfamily="monotype"; fontsize="16"; fontfill="blue"; stroke="blue"; a=text([-2.5,4],"x+2y=5",right); strokewidth=2; plot("0.5*(5-x)"); fontfill="red"; stroke="red"; b=text([2,3.5],"4x-2y=0",right); strokewidth=2; plot("2x"); fontfill="#990099"; stroke="#990099";dot([1,2],"closed","(1,2)",right); gg}
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(:div1end :)
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>>-<<
%b reg%Regulation:%% %b%A solution to the system is every intersection (touching) point of the 2 lines.
>>-<<
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[]>>&<<
(:showhide init=show div=div2 lshow="+" lhide="-":) %exa b%InterActivity%% (:showhide init=hide div=div21 lshow="+" lhide="-":) %sol s9%Directions for InterActivity%%
>>&<<
(:div2 id=div2 :)
>>&<<
(:div21 id=div21 class=s9 style="margin-left:20px":)
1. Look at the two lines %ref%a%% and %red%b%%. They intersect at the point %reg%E%%. Check that %reg%E%% satisfies both equations.
%s8 sol%If you know how, solve the system: %s8 ref%a%s8 sol% and %s8 red%b%s8 sol% and check that you get %s8reg%E.%%.
>>&<<2. Click and drag the lines or the points %ref%A%%, %ref%B%%, %red%C%% or %red%D%%. If they intersect at one point, %reg%E%% is "good".
%s8 sol%Look at the left (where the formulas are) to check that %s8 reg%E%s8 sol% is a point!%%
>>&<<3. Click and drag the points %ref%A%%, %ref%B%%, %red%C%% or %red%D%% so that %ref%a%% and %red%b%% are parallel. They are parallel when %reg%E%% says "undefined".
%s8 sol%Notice that %s8 ref%a%s8 sol% and %s8 red%b%s8 sol% are the same except for the constant after the "=" sign.%%
>>&<<4. Click and drag the points %ref%A%%, %ref%B%%, %red%C%% or %red%D%% so that %ref%a%% and %red%b%% coincide.
%s8 sol%Notice that %s8 ref%a%s8 sol% and %s8 red%b%s8 sol% are '''completely''' the same and that %s8 reg%E%s8 sol% is undefined.%%
>>-<<
(:div21end :)
>>&<<
(:div2end :)
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>>-<<
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[ colspan=3](:showhide init=hide div=div3 lshow="+" lhide="-":) Solutions to 2x2 linear systems
(:div3 id=div3 :)
%exa%A linear system %%
*The system has exactly one solution, i.e. the lines intersect in exactly one point and the solution is that point.
*The system has no solution, i.e. the lines are parallel and never touch.
*The system has infinitely many solutions, i.e. the lines coincide (are one and the same) and every point on this line is a solution.
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[]Exactly one solution: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right.$}
[]No solution: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x + 4y = 6} \\ \end{array} \right.$}
[]Infinitely many solutions: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x + 4y = 10} \\ \end{array} \right.$}
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[ width=32%]{gg width=240; height=180; xmin=-3; xmax=5; xscl=1; ymin=-1; ymax=5; yscl=1; axes(); fontfamily="monotype"; fontsize="16"; fontfill="blue"; stroke="blue"; a=text([-2.5,4],"x+2y=5",right); strokewidth=2; plot("0.5*(5-x)"); fontfill="red"; stroke="red"; b=text([2,3.7],"4x-2y=0",right); strokewidth=2; plot("2x"); fontfill="#990099"; stroke="#990099";dot([1,2],"closed","(1,2)",right); gg}
[ width=31%]{gg width=240; height=180; xmin=-3; xmax=5; xscl=1; ymin=-1; ymax=5; yscl=1; axes(); fontfamily="monotype"; fontsize="16"; fontfill="blue"; stroke="blue"; a=text([-2.5,4],"x+2y=5",right); strokewidth=2; plot("0.5*(5-x)"); fontfill="red"; stroke="red"; b=text([-.3,1.3],"2x+4y=6",left); strokewidth=2; plot("0.5*(3-x)"); gg}
[ width=37%]{gg width=240; height=180; xmin=-3; xmax=5; xscl=1; ymin=-1; ymax=5; yscl=1; axes(); fontfamily="monotype"; fontsize="16"; fontfill="blue"; stroke="blue"; a=text([-2.5,4],"x+2y=5",right); strokewidth=2; plot("0.5*(5.02-x)"); fontfill="red"; stroke="red"; b=text([-.3,2.1],"2x+4y=6",left); strokewidth=2; plot("0.5*(4.98-x)"); gg}
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(:div3end :)
>>&<<
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>>&<<
------
>>&<<
%rel%Related topics:
*&nbsp;
>>&<<
----
>>-<<
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