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(:title Good aschiiSVG graphs Systems of Linear Equations 2х2 - 2 equations in 2 variables:) >>nav_bar<<%reg%Mathcasts          %exa%Examples          %pra%Practice%%          [[ProblemsT/WindCurrent|%mor%More]]          %up%[[glossary/S]] >><< >>&<<[[<<]] >>-<< [table border=1 cellpadding=3 width=100%] [row] [l colspan=2] %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%%. [tableend] >>-<< [table border=1 cellpadding=3 width=100%] [row] [] (:showhide init=show div=div1 lshow="+" lhide="-":) Example of a linear system of 2 equations in 2 unknowns. (:div1 id=div1 :) [table border=1 cellpadding=3 cellspacing=0 width=100%] [row] [] Sample 2x2 linear system: {$ \left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right. $} >>&<>&< >>&<< {$ \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} [tableend] (:div1end :) [tableend] >>-<< %b reg%Regulation:%% %b%A solution to the system is every intersection (touching) point of the 2 lines. >>-<< [table border=1 cellpadding=3 width=100%] [row] []>>&<< (: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 :) (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="810" height="350" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/sistemi2a.ggb" showResetIcon="true" showToolBar="false" showMenuBar="false":) >>&<< (:div2end :) [tableend] >>-<< [table border=1 cellpadding=3 width=100%] [row] [ 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. [table border=1 cellpadding=3 cellspacing=0 width=100%] [row] []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. $} [row] [ 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} [tableend] (:div3end :) >>&<< [tableend] >>&<< ------ >>&<< %rel%Related topics: *  >>&<< ---- >>-<< [table width=100%] [row] [][[glossary/S| Attach:main/tri_purple_up_a.gif ]] [[glossary/S| Up one level]] [r](:html:) (:htmlend:) [tableend]