{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 277 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 276 41 " Solving 2 x 2 Systems of Linear Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Solving 2x2 systems of equations u sing three methods - graphing, substitution, and elimination" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 153 "[Directions : Execute the Code Resource section first. Although there will be no ou tput immediately, these definitions are used later in this worksheet.] " }{TEXT -1 1 "\n" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1575 "SysPlot := proc( eqns ) \n local eq1, eq2, sys,S,soln, x1,y1,xl,xr,L1,L2,H,V,ST,Point,wid, m1,m2;\n\n eq1 := eqns[1]:\n eq2 := eqns[2]:\n\n #------- \+ check for solubility ----------------\n m1 := subs(x = 1,solve(eq1 ,y))- subs(x = 0,solve(eq1,y));\n m2 := subs(x = 1,solve(eq2,y))- \+ subs(x = 0,solve(eq2,y)); \n if(m1<>m2) \n #------- solve t he system ---------------- \n then \n sys := \{eq1, \+ eq2\};\n y1 := solve( eq1, y);\n subs(y = y1, eq2);\n \+ x1 := solve( %, x);\n y1 := subs( x = x1, y1); \n s oln := [x1, y1]; \n xl := min( -1, 1.5*x1 ); xr := max( 1, 1.5* x1); \n wid := ((xr-xl)/20); \n \n #------- create pl ots ----------------\n Point := plottools[disk]([ x1, y1], wid, color=red ):\n H := plot( [[ 0,y1],[x1,y1]], color = orange, l inestyle = 2 );\n V := plot( [[x1,0], [x1,y1]], color = gold, \+ linestyle = 2 );\n ST := textplot( [ x1+1,y1-1, convert(soln,st ring) ] );\n L1 := plot( solve( eq1, y), x = xl..xr,thickness = 2, color=blue ):\n L2 := plot( solve( eq2, y), x = xl..xr,thic kness = 2, color=green ):\n plots[display]( L1, L2, Point, H, V , ST, scaling=constrained );\n\n #------- Display Result --------- ------- \n else \n xl:=-5; xr:=5;\n L1 := plot( sol ve( eq1, y), x = xl..xr,thickness = 2, color=blue ):\n L2 := plot( solve( eq2, y)+.08, x = xl..xr,thickness = 2, color=green ): \+ \n plots[display]( L1, L2, s caling=constrained);\n fi;\n \n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "Eliminate := proc( sys)\n print(`Add th ese two equations:`);\n print(sys[1]); print(sys[2]); \n print(`____ _______`); \n sys[1]+sys[2];\nend proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "1. Solving Systems Graphically" }}{PARA 0 "" 0 "" {TEXT -1 88 "\nA crude but insightful method of solving a system of equation s is by examining a graph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sys:= \{ 12*x + 13*y = 11, 7*x - 6*y = 20\};\nSysPlot ( sys );\nso lve( sys, \{x,y\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sys := \{ 2*x + 5*y = -7, 3*x - 6*y = -24\};\nsolve( sys, \{x,y\} );\nSysP lot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sys:= \{ 2* x + 7*y = 28, 3*x - 5*y = 11\};\nsolve( sys, \{x,y\} );\nSysPlot ( sys );" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "2. The Three Cases" }}{PARA 0 "" 0 "" {TEXT -1 149 "\nIn \+ the examples above, we were able to find a solution. More generally, w hen you attempt to solve a system, there are three possible outcomes : \n\n\011 " }{TEXT 256 1 "1" }{TEXT -1 89 ". The system is independent and consistent - in which case there is a single solution\n\011 " } {TEXT 257 1 "2" }{TEXT -1 69 ". The system is inconsistent - in which \+ case there is no solution\n\011 " }{TEXT 258 1 "3" }{TEXT -1 194 ". T he system is dependent - in which case there are infinite number of s olutions.\n\nLets look what happens when we encounter these two later \+ types of systems which don\324t yield a single solution." }}{PARA 0 " " 0 "" {TEXT -1 1 "\n" }}{PARA 257 "" 0 "" {TEXT -1 36 "I. Independent & Consistent Systems " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 205 "Here is an independent and consistent system. Its c onsistent because the lines intersect, and the lines are independent - meaning that they are not the same line. Any two non-parallel lines a re consisent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sys:= \{ 2*x + 5*y = 28, 3*x - 6*y = 15\};\nsolv e( sys, \{x,y\} );\nSysPlot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sys:= \{ 8*x + 3*y = 14, 11*x + 2*y = -2\};\nsolve( s ys, \{x,y\} );\nSysPlot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 257 "" 0 "" {TEXT -1 25 "II. Inconsistent Systems " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "Here is an inconsistent system. T he system is not consistent because the lines do not intersect. The li nes are parallel and Maple is unable to solve the system since there i s no solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sys:= \{ 3*x + 7*y = 42, 3*x + 7*y = 0 \};\nSysP lot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sys:= \{ 10 *x - 2*y = 40, 10*x - 2*y = 20 \};\nSysPlot ( sys );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "\n" }}{PARA 257 "" 0 "" {TEXT -1 24 "III. Dependent Systems " }}{PARA 0 "" 0 "" {TEXT -1 327 "\nHere is an example of dependent system. If two lines are actual ly the same line, they are called dependent. There is only one line be cause the two lines in the system are actually one and the same. In th is case, EVERY point is a point of intersection. There are an infinite number of solutions - all of the points on THE line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sys := \+ \{ 4*x - 6*y = 30, 3*y = 2*x-15 \};\nSysPlot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sys := \{ y = -2*x + 12, 2*y = 24 \+ -4*x \};\nSysPlot ( sys );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 47 "3. Solving Systems Algebraically - Substitution" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 259 13 "Problem 3.1 :" }{TEXT -1 65 " 3x + 8y = -1 \n 5x - 2y = 29\n\n " }{TEXT 260 2 "0." }{TEXT -1 17 " Write the system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sys := \{ 3*x + 8*y = -1 , 5*x - 2*y = 2 9 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 7 " \n " }{TEXT 261 2 "1." } {TEXT -1 302 " Solve one of the equations for one of the variables. Th ere are four choices, but choose the one that \n looks easiest. In this case, they all look to be roughly the same level of difficult y. Lets solve for y\n in the first equation. Note that y will be expressed in terms of numbers and x.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y1 := solve( sys[1], y );" }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n " }{TEXT 262 2 "2." }{TEXT -1 197 " Now substitute THIS y, for the y in the OTHER equation. This is the key to this method. Once we \n do that, we get an equation - but only in one variable x . This is an equation we can solve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n " }{TEXT 263 2 "3." }{TEXT -1 26 " Solve this equation for x" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }}} {PARA 0 "" 0 "" {TEXT -1 5 "\n " }{TEXT 264 2 "4." }{TEXT -1 62 " S ubstitute this x value back into EITHER equation, to find y." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1 := subs( x = x1, y1);" }} }{PARA 0 "" 0 "" {TEXT -1 27 "\n Here is the solution! " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "soln := [x1, y1];" }}}{PARA 0 "" 0 "" {TEXT -1 179 " \n We can check that this solution is correct \+ by substituting it back into both equations. Note that both\n equa tions become true statements such as -1 = -1. We are done." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( \{x = x1, y = y1\}, \{sys[1], sys[2]\});" }}}{PARA 0 "" 0 "" {TEXT -1 101 " Using Maple, there is another way to check our answer - simply solve using a single Maple c ommand." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( sys, \{x,y \} );" }}}{PARA 0 "" 0 "" {TEXT -1 35 "\n\n\n\nLets try some other exa mples.\n\n" }{TEXT 275 13 "Problem 3.2 :" }{TEXT -1 21 " Solve the sys tem : " }{XPPEDIT 18 0 "7*x+2*y = 8;" "6#/,&*&\"\"(\"\"\"%\"xGF'F'*& \"\"#F'%\"yGF'F'\"\")" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "5*x-2*y = 16;" "6#/,&*&\"\"& \"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'!\"\"\"#;" }{TEXT -1 72 "\n\nIf we add \+ these equations together, none of the variables will cancel." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sys := \{ 7*x + 2*y = 8 \+ , 5*x - 2*y = 16 \}:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1 := solve( sys[1], y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " y1 := subs( x = x1, y1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "soln := [x1, y1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 " \n We can check that this solution is correct b y substituting it back into both equations or using the Maple command. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( \{x = x1, y = y1\} , \{sys[1], sys[2]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s olve( sys, \{x,y\} );" }}}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}{PARA 0 " " 0 "" {TEXT -1 29 "\n\nLets try another example.\n\n" }{TEXT 273 13 " Problem 3.3 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "11 *x+6*y = 8;" "6#/,&*&\"#6\"\"\"%\"xGF'F'*&\"\"'F'%\"yGF'F'\"\")" } {TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "5*x-3*y = 16;" "6#/,&*&\"\"&\"\"\"%\"xGF'F'*&\"\"$F' %\"yGF'!\"\"\"#;" }{TEXT -1 72 "\n\nIf we add these equations together , none of the variables will cancel." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys := \{ 11*x + 6*y = 10 , 5*x - 3*y = 16 \}: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1 := solve( sys[1], y) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[2]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y1 := subs( x = x1, y1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "soln := [x1, y1];" }}} {PARA 0 "" 0 "" {TEXT -1 9 "\n\nCheck :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( sys, \{x,y\} );" }}}{PARA 0 "" 0 "" {TEXT -1 40 "\n\n\n\n\nWhat if we get a dependent system?\n" }{MPLTEXT 1 0 1 " \n" }{TEXT 274 13 "Problem 4.6 :" }{TEXT -1 21 " Solve the system : \+ " }{XPPEDIT 18 0 "15*x-10*y = 45;" "6#/,&*&\"#:\"\"\"%\"xGF'F'*&\"#5F' %\"yGF'!\"\"\"#X" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "-18*x+12*y = -54;" "6#/,&*&\"#=\" \"\"%\"xGF'!\"\"*&\"#7F'%\"yGF'F',$\"#aF)" }{TEXT -1 15 "\n\nSometimes w\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys := [ 15*x - 1 0*y = 45, -18*x + 12*y = -54 ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1 := solve( sys[1], y);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "subs(y = y1, sys[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 311 "\nWe can't do anything with this resulting equation because it ha s no variables. However, this equation is a true statement. This means our assumption that the lines have a point of intersection is true .. . true for all values of x. Therefore this is a depedent system and th e solution is all points on the line." }}{PARA 0 "" 0 "" {TEXT -1 76 " \n\n\nIf we attempt the substitution method on an inconsistent system, we get :" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 272 13 "Problem \+ 4.7 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "15*x-10*y \+ = 45;" "6#/,&*&\"#:\"\"\"%\"xGF'F'*&\"#5F'%\"yGF'!\"\"\"#X" }{TEXT -1 55 "\n " } {XPPEDIT 18 0 "-18*x+12*y = 54;" "6#/,&*&\"#=\"\"\"%\"xGF'!\"\"*&\"#7F '%\"yGF'F'\"#a" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sys := [ 15*x - 10*y = 45, -18*x + 12*y = 54 ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1 := solve( sys[1], y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[2]);" }}} {PARA 0 "" 0 "" {TEXT -1 324 "\nClearly something is wrong. We can't s olve for any variable, and we now have a FALSE statement. It must be t he case, that there is no point of intersection. In other words, we ha ve an inconsistent sytem - parallel, non-intersecting lines - so the a nwer to this problem is the empty set. There is no point of intersecti on.\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "4. Solving Systems Algebraically - Elimination" }}{PARA 0 "" 0 "" {TEXT -1 71 " \nSometimes we get lucky. Lets just add these \+ two equations together.\n\n" }{TEXT 266 13 "Problem 4.1 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "7*x+2*y = 8;" "6#/,&*&\"\"( \"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'F'\"\")" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "5*x - 2*y = \+ 16" "6#/,&*&\"\"&\"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'!\"\"\"#;" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sys := \{ 7*x + 2 *y = 8 , 5*x - 2*y = 16 \}:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Eliminate( [sys[1], sys[2]] );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(x = x1, sys[1]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "y1 := solve( %, y);\n[x1, y1];\nsolve( sys, \{x,y\} );" }}}{PARA 0 "" 0 "" {TEXT -1 173 "\nThis worked out well because we were lucky to have two terms which cancelled : 2y and -2y. What if we are not quite so lucky? A little cleverness will bring luck our way. \n\n" }{TEXT 267 13 "Problem 4.2 :" }{TEXT -1 21 " Solve the system : \+ " }{XPPEDIT 18 0 "11*x+6*y = 8;" "6#/,&*&\"#6\"\"\"%\"xGF'F'*&\"\"'F' %\"yGF'F'\"\")" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "5*x-3*y = 16;" "6#/,&*&\"\"&\"\"\"% \"xGF'F'*&\"\"$F'%\"yGF'!\"\"\"#;" }{TEXT -1 72 "\n\nIf we add these e quations together, none of the variables will cancel." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys := \{ 11*x + 6*y = 10 , 5*x - 3* y = 16 \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Eliminate( [ sys[1], sys[2]] );" }}}{PARA 0 "" 0 "" {TEXT -1 104 "\n\nHowever, if w e add the first equation to TWO TIMES the second equation, then it wor ks out much better." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Elimi nate( [sys[1], 2*sys[2]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " subs(x = x1, sys[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "y1 := solve( %, y);\n[x1, y1];\nsolve( sys, \{x,y\});" }}}{PARA 0 "" 0 " " {TEXT -1 152 "\nWhy are we justified in multiplying an equation by a constant? We did the exact same thing earlier, when were solving equa tions. When we want to solve " }{XPPEDIT 18 0 "x/7 = 13" "6#/*&%\"xG\" \"\"\"\"(!\"\"\"#8" }{TEXT -1 632 ", we multiply both sides by 7. In f act, we added numbers to both sides of equations, subtracted numbers f rom both sides of an equation, multiplied both sides, and divided both sides. So we have a long history of performing the same operation to \+ both sides of an equation, and we know this will not change the nature and quality of the equation. \n\nNow we are doing it to cause two var iables to cancel. We are trying to find a number to multiply both side s of an equation by, so that one of the varibles cancel. In the last c ase, we were somewhat lucky because 2 goes into 6.\n\n\nHere is anothe r example. This time, we'll eliminate x.\n\n" }{TEXT 268 13 "Problem 4 .3 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "7*x+9*y = 2 3;" "6#/,&*&\"\"(\"\"\"%\"xGF'F'*&\"\"*F'%\"yGF'F'\"#B" }{TEXT -1 55 " \n " }{XPPEDIT 18 0 "x-8*y = -6;" "6#/,&%\"xG\"\"\"*&\"\")F&%\"yGF&!\"\",$\"\"'F*" } {TEXT -1 71 "\n\nSometimes we get lucky. Lets just add these two equat ions together.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sys := \{ 7*x - 9*y = 23 , x + 8*y = -6 \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Eliminate( [sys[1], -7*sys[2]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y1 := solve( %, y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x1 := solve( %, x);\n[x1, y1];\nsol ve( sys, \{x,y\});" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 287 "\nWhat if we are completely out of luck? Then we'll h ace to manufacture some luck. If there is no integer we can multiply t o make a variable cancel, then we multiply both equations to transform them both into equations which have equal but opposite coefficients o n one of the variables.\n\n" }{TEXT 269 13 "Problem 4.4 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "3*x+7*y = 1;" "6#/,&*&\"\"$ \"\"\"%\"xGF'F'*&\"\"(F'%\"yGF'F'F'" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "4*x-5*y = 30;" "6#/,&*&\"\"%\"\"\"%\"xGF'F'*&\"\"&F'%\"yGF'!\"\"\"#I" }{TEXT -1 2 "\n \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sys := \{ 3*x + 7*y \+ = 1 , 4*x - 5*y = 30 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 234 "\nLets \+ first decide which variable we wish to get rid of. Lets choose x. In t he first equation, we have 3x and in the 2nd equation we have 4x. What is the smallest number which both 3 and 4 go into? This seems like a \+ familar situation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "lcm(3,4 );" }}}{PARA 0 "" 0 "" {TEXT -1 88 "\nSo we will need to multiply the \+ first equation by 12/3 = 4 and the second by 12/4 = 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Eliminate( [4*sys[1], 3*sys[2]] ); " }}}{PARA 0 "" 0 "" {TEXT -1 225 "\nOops. That didn't work out! We st ill have both variables! We want these multiplied equations to cancel \+ out their x terms, so lets multiply the second one by -3 rather than + 3. Then we'll get +12x and -12x, which will cancel." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Eliminate( [4*sys[1], -3*sys[2]] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y1 := solve( %, y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y = y1, sys[1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x1 := solve( %, x);\n[x1, y1 ];\nsolve( sys, \{x,y\});" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 29 "\n\nHere is another example :\n\n" }{TEXT 270 13 "Problem 4.5 :" }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "7*x+2*y = 8;" "6#/,&*&\"\"(\"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'F'\"\")" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "5*x - 2*y = 16" "6#/,&*&\"\"&\"\"\"%\"xGF'F'*&\"\"#F '%\"yGF'!\"\"\"#;" }{TEXT -1 20 "\n\nLets eliminate y.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sys := \{ 7*x + 6*y = 9 , 5*x \+ + 8*y = 25 \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lcm(6,8) ; %/6; %%/8;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Eliminate ( 4*[sys[1], 1*sys[2]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1 := solve( %, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " subs(x = x1, sys[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "y1 := solve( %, y);\n[x1, y1];\nsolve( sys, \{x,y\});" }}}{PARA 0 "" 0 " " {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 36 "\nWhat if we get a \+ dependent system?\n" }{MPLTEXT 1 0 1 "\n" }{TEXT 271 13 "Problem 4.6 : " }{TEXT -1 21 " Solve the system : " }{XPPEDIT 18 0 "15*x-10*y = 45; " "6#/,&*&\"#:\"\"\"%\"xGF'F'*&\"#5F'%\"yGF'!\"\"\"#X" }{TEXT -1 55 " \n " }{XPPEDIT 18 0 "-18*x+12*y = -54;" "6#/,&*&\"#=\"\"\"%\"xGF'!\"\"*&\"#7F'%\"yGF' F',$\"#aF)" }{TEXT -1 15 "\n\nSometimes w\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys := [ 15*x - 10*y = 45, -18*x + 12*y = -54 \+ ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lcm( -10, 12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Eliminate( [6*sys[1], 5*sys[ 2]] );" }}}{PARA 0 "" 0 "" {TEXT -1 402 "\nThat pretty much stops us i n our tracks. We can't solve this resulting equation for x or y. Howev er, this equation is a true statement. This means our assumption that \+ the lines have a point of intersection is true ... true for all values of x. Therefore this is a depedent system and the solution is all poi nts on the line.\n\n\n\n\n\nIf we attempt the elimination method on an inconsistent system, we get :" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" } {TEXT 265 13 "Problem 4.7 :" }{TEXT -1 21 " Solve the system : " } {XPPEDIT 18 0 "15*x-10*y = 45;" "6#/,&*&\"#:\"\"\"%\"xGF'F'*&\"#5F'%\" yGF'!\"\"\"#X" }{TEXT -1 55 "\n \+ " }{XPPEDIT 18 0 "-18*x+12*y = 54;" "6#/,&*&\"#=\"\"\"% \"xGF'!\"\"*&\"#7F'%\"yGF'F'\"#a" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sys := [ 15*x - 10*y = 45, -18*x + 12*y = 5 4 ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lcm( -10, 12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Eliminate( [6*sys[1], 5*sys[ 2]] );" }}}{PARA 0 "" 0 "" {TEXT -1 322 "\nClearly something is wrong. We can't solve for any variable, and we now have a FALSE statement. I t must be the case, that there is no point of intersection. In other w ords, we have an inconsistent sytem - parallel, non-intersecting lines - so the anwer to this problem is the empty set. There is no point of intersection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 279 36 "\n \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 32 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }