(:title Systems of Linear Equations 2х2 - 2 equations in 2 variables:) >>nav_bar<<[[Glossary/Systems2mc|%reg%Mathcasts]]          %exa%Examples          %pra%Practice%%          [[InterA/BoatCurrent|%mor%More]]          %up%[[glossary/S]] >><< >>&<<[[<<]] >>-<< [table border=1 cellpadding=3 width=825] [row] [l colspan=2] %def b%Definition%%: Finding a simultaneous solution to 2 linear equations in 2 variables - 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=825] [row] [] (:showhide init=show div=div1 lshow="+" lhide="-":) Example of a linear system of 2 equations in 2 variables (:div1 id=div1 :) [table border=1 cellpadding=3 cellspacing=0 width=100%] [row] [] Sample 2x2 linear system: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} >>&<>&<<%pra%Why is this a solution?%% Because - when we >>&<< -> substitute {$\color{purple}{x=1}$} and {$\color{teal}{y=2}$}, both equations are "true". --> >>&<< {$\begin{array}{c} \color{purple}{1}\color{blue}{+}\color{teal}{2}\color{blue}{ \cdot 2 \,\mathop = \limits^?\, 5} \\ \color{blue}{1 + 4 \,\mathop = \limits^?\, 5} \\ \color{blue}{5 \,=\, 5} \\ \end{array}$} {$\begin{array}{c} \color{red}{2\cdot} \color{purple}{1} \color{red}{- } \color{teal}{2}\color{red}{ \,\mathop = \limits^?\, 0} \\ \color{red}{2 - 2 \,\mathop = \limits^?\, 0} \\ \color{red}{0 = 0} \\ \end{array}$} [c width=300]Graphically, we have: >>&<< (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="280" height="180" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/sistemi2exa.ggb" showResetIcon="true" showToolBar="false" showMenuBar="false":) [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=825] [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=825] [row] [ colspan=3](:showhide init=show div=div3 lshow="+" lhide="-":) Solutions to 2x2 linear systems (:showhide init=hide div=div312 lshow="+" lhide="-":) %b red%Graphs!%% (:div3 id=div3 :) %exa%A linear system can have: %% (:div 311 class=s9:) [table cellspacing=1 cellpadding=3 width=100%] [row bgcolor=#EEFFEE] [] • Exactly %b%one solution%%. The lines intersect in exactly one point. %sol s9%Examples below in "Solution Methods".%% >>&<< --> When solving this system you get numbers for x and y. %b pra%Answer is: (%b mor%number for x%b pra%, %b bgr%number for y%b pra%)%% [row bgcolor=#FFDDDD] [] • %b%No solution%%. The lines are parallel and never touch. The system is [[GlossaryT/SystemsInconsistent|%mor%inconsistent%%]]. >>&<< --> When solving this system you get something stupid like 3=5. %b red%Answer is: No solution.%% [row bgcolor=#EEEEEE] [] • %b%Infinitely many solutions%%. The lines coincide. They are the same line. Every point on this line is a solution. [[GlossaryT/SystemsInfinite|%mor%More%%]] >>&<< --> When solving this system you get stuck with %ref%0=0%%. %b dkr%Answer is: Many solutions.%% [tableend] (:div311end:) >>-<< (:div312 id=div312 :) [table border=1 cellpadding=3 cellspacing=0 width=100%] [row] [c]Exactly one solution: %b%(1,2)%% {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} [c]No solution {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{x + 2y = 2} \\ \end{array} \right.$} [c]Infinitely many solutions {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x + 4y = 10} \\ \end{array} \right.$} >>&<< [row] [c](:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="260" height="180" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/sistemi2exa.ggb" showToolBar="false" showMenuBar="false":) [c](:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="260" height="180" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/sistemi2exb.ggb" showToolBar="false" showMenuBar="false":) [c](:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="260" height="180" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/sistemi2exc.ggb" showToolBar="false" showMenuBar="false":) [tableend] (:div312end :) (:div3end :) >>&<< [tableend] >>-<< [table border=1 cellpadding=3 width=825] [row] [](:showhide init=show div=div5 lshow="+" lhide="-":) Solution methods >>&<< (:div5 id=div5 :) %exa%A linear system can be solved using any of the following three methods: %% *The %b exa%substitution%% method. (:showhide init=hide div=div511 lshow="+" lhide="-":) %b exa%Example%% *The %b pra%addition%% or elimination method (:showhide init=hide div=div512 lshow="+" lhide="-":) %b pra%Example%% *Cramer's rule (%b mor%determinants%%) (:showhide init=hide div=div513 lshow="+" lhide="-":) %b mor%Example%% (:div51 id=div51 class=s9:) >>-<< (:div511 id=div511:) [table border=1 cellpadding=3] [row] []The %b exa%substitution%% method. Solve: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} [table border=0 cellpadding=3] [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{blue}{x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{blue}{x = 5-2y} \\ \color{navy}{2x - y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{2\cdot\color{blue}{(5-2y)} - y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{10-4y - y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{10-5y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{10=5y} \\ \end{array} \right.$} []{$\Leftrightarrow$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{y=2} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{5y=10} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{blue}{x = 5-2 \cdot} \color{red}{2} \\ \color{red}{y=2} \\ \end{array} \right.$} []{$\Leftrightarrow$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{blue}{x = 5-4} \\ \color{navy}{y=2} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{blue}{x = 1} \\ \color{navy}{y=2} \\ \end{array} \right.$} []  [ colspan=2]Solution is: (1,2) [tableend] [c valign="middle"]%popwin width=500 height=440%[[http://mathcasts.org/gg/enliven/af/systems/substitute2x2/substitute2x2.htm|Attach:systems2x2_sub.jpg]] >>&<<[[Glossary/Systems2mc|More Mathcasts]] [tableend] >>&<< (:div511end:) (:div512 id=div512 :) [table border=1 cellpadding=3] [row] []The %b pra%addition%% method. Solve: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} [table border=0 cellpadding=3] [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{blue}{x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x + 2y = 5} \\ \color{purple}{5x+0y =5} \\ \end{array} \right.$} []{$\Leftrightarrow$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{navy}{x + 2y = 5} \\ \color{purple}{5x=5} \\ \end{array} \right.$} [] {$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{navy}{x + 2y = 5} \\ \color{purple}{x=1} \\ \end{array} \right.$} [] {$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{purple}{1} \color{blue}{+2y=5} \\ \color{navy}{x=1} \\ \end{array} \right.$} []{$\Leftrightarrow$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$\left\{ \begin{array}{l} \color{blue}{2y=4} \\ \color{navy}{x=1} \\ \end{array} \right.$} []{$\Leftrightarrow$} []{$\left\{ \begin{array}{l} \color{blue}{y=2} \\ \color{navy}{x=1} \\ \end{array} \right.$} []  [ colspan=2]Solution is: (1,2) [tableend] [c valign="middle"]%popwin width=500 height=440%[[http://mathcasts.org/gg/enliven/af/systems/addition2x2/addition2x2.htm|Attach:systems2x2_add.jpg]] >>&<<[[Glossary/Systems2mc|More Mathcasts]] [tableend] >>&<< (:div512end:) (:div513 id=div513 :) [table border=1 cellpadding=3] [row] []%b mor%Cramer's rule%%. Solve: {$\left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x - y = 0} \\ \end{array} \right.$} = {$\left\{ \begin{array}{c} \color{blue}{1}x + \color{#FF8000}{2}y = \color{orange}{5} \\ \color{red}{2}x \color{#800080}{ - 1} y = \color{#00BB80}{0} \\ \end{array} \right.$} [table border=0 cellpadding=3] [row] [ colspan=6]------------------------------------------------------------------------------- [row] [colspan=6]{$D = \left| {\matrix{ \color{blue}{1} & \color{#FF8000}{2} \cr \color{red}{2} & \color{#800080}{ - 1} \cr } } \right| = \color{blue}{1} \cdot \color{#800080}{ ( - 1)} - \color{red}{2} \cdot \color{#FF8000}{2} = - 1 - 4 = - 5$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] [colspan=6]{$\color{purple}{D_x} = \left| {\matrix{ \color{orange}{5} & \color{#FF8000}{2} \cr \color{#00BB80}{0} & \color{#800080}{ - 1} \cr } } \right| = \color{orange}{5} \cdot \color{#800080}{ ( - 1)} - \color{#00BB80}{0} \cdot \color{#FF8000}{2} = - 5 - 0 = - 5$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] [colspan=6]{$\color{teal}{D_y} = \left| {\matrix{ \color{blue}{1} & \color{orange}{5} \cr \color{red}{2} & \color{#00BB80}{0} \cr } } \right| = \color{blue}{1} \cdot \color{#00BB80}{0} - \color{red}{2} \cdot \color{orange}{5} = 0 - 10 = - 10$} [row] [ colspan=6]------------------------------------------------------------------------------- [row] []{$x=\frac{\,D_x\,}{D}=\frac{-5}{-5}=1$} []  []{$y=\frac{\,D_y\,}{D}=\frac{-10}{-5}=2$} []  [ colspan=2]Solution is: (1,2) [tableend] [] [tableend] >>&<< (:div513end :) (:div51end:) (:div5end :) >>&<< [tableend] >>&<< [table border=1 cellpadding=3 width=825] [row] []>>&<<(:showhide init=hide div=div9 lshow="+" lhide="-":) Metadata (:div9 id=div9 :) [table class=column border=1 width=100% cellpadding=3] [row] [ width=110px] '''Global''' []2x2 System of Equations [row] []Brief []Mathcasts and Interactivities to understand and practice solving systems of 2 equations in 2 variables (unknowns). [row] []Grade []7-10    Interactivities start at 8th grade level on up [row] []Strand []Algebra; Expressions, Equations and Inequalities [row] []Standard []%popwin%[[standards/Is1al3-1?action=popopen|Algebra 1 3.1]], %popwin%[[standards/Is1al3-2?action=popopen|Algebra 1 3.2]], %popwin%[[standards/Is1al3-3?action=popopen|Algebra 1 3.3]], %popwin%[[standards/ACTee28?action=popopen|ACT EE 28-32]] [row] []Keywords []system, linear system, variable, unknown, linear equation, elimination, addition, substitution [row] []Comments []none [row] []Download []  [row] []Author []LFS - [[mailto:emath@emathforall.com?subject=ggb_slopes|contact]] - [[http://math247.pbwiki.com/Algebra-with-GeoGebra|website]] [row] []Type []Freeware - Available for Offline and Online Use - Translatable (html) [row] []Use []Requires [[http://www.java.com/en/download/index.jsp|sunJava player]] [tableend] (:div9end:) [tableend] >>&<< ------ >>&<< %rel%Related topics: *[[Lines]] *[[GlossaryT/SystemsInconsistent|Inconsistent Systems of Linear Equations]] *[[GlossaryT/LinearFunction|Linear Functions]] *[[GlossaryT/Slope|Slope of a Line]] *[[GlossaryT/LinearEquation|Linear Equations]] >>&<< ---- >>-<< [table width=100%] [row] [][[glossary/S| Attach:main/tri_purple_up_a.gif ]] [[glossary/S| Up one level]] [r](:html:) (:htmlend:) [tableend] [[!Algebra 1]] [[!Linear Systems]] [[!Is1al3-1]] [[!Is1al3-2]] [[!Is1al3-3]] [[!ACTee28]]