二元一次方程式與直角坐標平面的關係

假設有兩條線過那一點且方程式化簡後仍然不一樣，則他們的斜率不一樣. 接著又發現如果把方程式寫成y=ax+b的形式，a就是斜率. 假設那條線上其中兩點的坐標為(x1,ax1+b) ... ¤G¤¸¤@¦¸¤èµ{¦¡»Pª½¨¤§¤¼Ð¥­­±ªºÃö«Y ¦ó¿×¤èµ{¦¡ªº¸Ñ? ¸Ñªº·N«ä´N¬O¥N¤J·|º¡¨¬¤èµ{¦¡ ¨Ò¦p:¤èµ{¦¡2x+3y=5ªº¨ä¤¤¤@­Ó¸Ñ¬°x=1,y=1¡A¦]¬°2*1+3*1=5 ¤G¤¸¤@¦¸¤èµ{¦¡¦bª½¨¤§¤¼Ð¥­­±¤Wªº¹Ï§Î ¦]¬°¤G¤¸¤@¦¸¤èµ{¦¡ªº¸Ñ³£¬Ox=?,y=?¡A©ó¬O§âxªº­È¸òyªº­È¥Î§¤¼Ð¨Óªí¥Ü ¨Ò¦p:¤W­±ªº¨Ò¤l¤¤±o2x+3y=5ªº¤@¸Ñ¬°x=1,y=1¡A¥Î§¤¼Ðªí¥Ü¬°(1,1)¡A¦A§â§¤¼Ð¼Ð¦b§¤¼Ð¥­­±¤W §â¤èµ{¦¡ªº¸Ñ³£¼Ð¦b§¤¼Ð¥­­±¤W´N¬O¤èµ{¦¡ªº¹Ï§Î¤F §Ú­Ì²ßºD¤W·|¥Îªí®æ¨Ó°O¿ý¤èµ{¦¡ªº¸Ñ ¦p2x+3y=5ªº¨ä¤¤¤­²Õ¸Ñ¬° ±N¥L­Ìµe¦b§¤¼Ð¥­­±¤W ±qx=-5¨ìx=5¶¡§ä100­ÓÂI«á¼Ð¦b§¤¼Ð¥­­±¤Wªº¹Ï§Î ¦pªG§â2x+3y=5ªº¹Ï§Îµe¦b§¤¼Ð¥­­±¤W¬° ¦]¦¹§Ú­Ìµo²{§â¤G¤¸¤@¦¸¤èµ{¦¡µe¦bª½¨¤§¤¼Ð¥­­±¤W¬Oª½½u ¤S§Ú­Ìª¾¹D¨âÂI´N¥i¥Hµe¤@±øª½½u¡A©Ò¥H¥u­n§ä¤èµ{¦¡ªº¨â²Õ¸Ñ´N¥i¥Hµe¤èµ{¦¡ªº¹Ï§Î¤F ¦p¦ó´N±ø¥ó¨Ó§ä¥X"ª½½u"¤èµ{¦¡ ¦pªGµ¹©w¨âÂI­n§ä¤èµ{¦¡(¦]¬°¨âÂI¥iµe¥X¤@½u¡A¤Sª½½u³£¥i¥H§ä¨ì»P¨ä¹ïÀ³ªº¤G¤¸¤@¦¸¤èµ{¦¡) ²Ä¤@ºØ¤è¦¡¥i¥H°²³]¤èµ{¦¡¬°y=ax+b(¤£¥i¥H°²³]ax+by=c¡A¦]¬°¤T­Ó¥¼ª¾¼Æ»Ý­n¤T­Ó¦¡¤l¤~¯à¸Ñ¡A¦Ó¨âÂI¥u¦³¨â­Ó¦¡¤l)¡A¦A§â¨âÂI¥N¤J¨D¥Xa©Mb ¨Ò¦p­n¨D¹L(1,1)©M(-2,3)ªºª½½u¤èµ{¦¡ ­º¥ý¥ý°²³]¤èµ{¦¡¬°y=ax+b ¦A¨Ó±NÂI¥N¤J±o1=a*1+b©M3=a*(-2)+b¡A±µµÛ¸ÑÁp¥ß´N¥i¥H±o¥Xa=-2/3,b=5/3 ©Ò¥H¹L¨º¨âÂIªº¤èµ{¦¡¬°y=(-2/3)x+5/3 ±µµÛ¤¶²ÐÂI±×¦¡¡A­º¥ý¤¶²Ð±×²v ±×²v´N¦p¦P¥¦¦r­±ªº·N«ä¡A§Y¶É±×ªº¤ñ²v ¥i¥H¥Î¤W¤U¼Ó±èªº·Qªk¨Ó¬Ý ¶V±×ªº·N«ä´N¬O¶VÃø¨«¡A§Y¨C¤@¨B­n¨«ªº°ª«×¶V°ª ©Ò¥H±×²v¤]´N¬Ox¼W¥[1®Éy¼W¥[ªº­È ·íµM¼Ó±è¤]¦³¦V¤Uªº ·í¦V¤U®É±×²v´N¬O­tªº §Ú­Ì·|¥Î¨º±ø½u¤W¥ô¨âÂIªºy­È®t°£¥Hx­È®t¨Óºâ¨º±ø½uªº±×²v *ª`·N¦pªGx­È®t¬O¥ÎaÂIªºx­È´îbÂIªºx­È«hy­È®t¤]­n¥ÎaÂIªºy­È¥h´îbÂIªºy­È ¦Ó¥B¦pªG¹L¤@ÂI¥B±×²v©T©w«h¥u·|¦³¤@±ø½uº¡¨¬ °²³]¦³¨â±ø½u¹L¨º¤@ÂI¥B¤èµ{¦¡¤Æ²«á¤´µM¤£¤@¼Ë¡A«h¥L­Ìªº±×²v¤£¤@¼Ë ±µµÛ¤Sµo²{¦pªG§â¤èµ{¦¡¼g¦¨y=ax+bªº§Î¦¡¡Aa´N¬O±×²v °²³]¨º±ø½u¤W¨ä¤¤¨âÂIªº§¤¼Ð¬°(x1,ax1+b),(x2,ax2+b) «h¨º±ø½uªº±×²v¬°((ax1+b)-(ax2+b))/(x1-x2)¡A·|µ¥©óa ©Ò¥H¤]¥i¥H§Q¥Î±×²v¨Ó¨D¤èµ{¦¡ ¦Ó¥Ñ±×²v©MÂI§ä¥X¤èµ{¦¡ªº¤è¦¡´NºÙ¬°ÂI±×¦¡ ¦¡¤l¬°y-b=m(x-a)©ÎªÌ¬°y=m(x-a)+b¨ä¤¤m¬O±×²v¡A(a,b)¬O³q¹LªºÂI(¦]¬°(a,b)¥N¤Jº¡¨¬¤èµ{¦¡) ¹³¬O¤W­±¨º­Ó¨Ò¤l ¥i¥H¥ýºâ±×²v ¥Îy­È®t°£¥Hx­È®t¨Ó¨D¡A¬°(1-3)/(1-(-2))=-2/3 ¦A¥Î¨ä¤¤¤@ÂI¸ò±×²v¨Ó§ä¥X¤èµ{¦¡±oy=-2/3(x-1)+1©Î¬Oy=-2/3(x-(-2))+3 ¤Æ²«á·|µo²{¨â¤èµ{¦¡¤@¼Ë Åý¦U¦ì¦Û¤v¸Õ¸Õ¬Ý¡A¦b®Ø®Ø¤¤¿é¤J±×²v¡A¥H¤Î§âÂI²¾¨ì³ßÅwªº¦ì¸m ³o­Ó¤pµ{¦¡¬O§Q¥ÎGeoGebra°µ¥X¨Óªº¡]©xºô¡Gwww.geogebra.org¡^¡A¥¦»Ý­n¥Î¨ìJava¡A¦ý±z¨S¦³¦w¸ËJava¡A½Ð¦Ü¡Gwww.java.com¦w¸Ë¡C GeoGebra製作 ¦A¨Ó¤¶²ÐºI¶Z¦¡ xºI¶Z´N¬Oy=0¥N¤J¤èµ{¦¡©Ò¨D¥Xªºx­È¡AyºI¶Z´N¬Ox=0¥N¤J¤èµ{¦¡©Ò¨D¥Xªºy­È ¦ÓºI¶Z¦¡´N¬O¥ÎºI¶Z¨Ó§ä¥X¤èµ{¦¡¡A¦¡¤l¬°x/a+y/b=1¨ä¤¤a¬°xºI¶Z¡Ab¬°yºI¶Z ¦]¬°·íy=0¥N¤J®É±ox=a¡Ax=0¥N¤J®É±oy=b *ª`·Nª½½uªºxºI¶Z¬°ª½½u¸òx¶bªº¥æÂI¨ìy¶bªº¶ZÂ÷¡AyºI¶Z¬°ª½½u¸òy¶bªº¥æÂI¨ìx¶bªº¶ZÂ÷ ¨â±ø½u¦bª½¨¤§¤¼Ð¥­­±ªºÃö«Y ¨â±ø½u¦b¥­­±¤WªºÃö«Y¥u¦³¥­¦æ¡B­«Å|¡B¥æ¤@ÂI¡B««ª½¨ä¤¤­«Å|ºâ¥­¦æªº¯S¨Ò¡A««ª½ºâ¥æ¤@ÂIªº¯S¨Ò ¥­¦æ ­«Å| ¥æ¤@ÂI ««ª½ ¤°»ò®É­Ô¥­¦æ©O? ·í¨â¤èµ{¦¡±×²v¬Û¦P®É¡A¨â±ø½u´N·|¥­¦æÅo ¦p2x+3y=5©M4x+6y=5¡A2/4=3/6 ­«Å|´N¬O¨â­Ó¤èµ{¦¡¨ä¹ê¬O¤@¼Ëªº¡A´N¬O¥i¥H§Q¥Îµ¥¶q¤½²z§â¨â±ø¤èµ{¦¡§Ë¦¨¤@¼Ë ¦p2x+3y=5©M4x+6y=10¡A2(2x+3y)=2*5¡÷4x+6y=10 ­Y¨â±ø½u¤£¥­¦æ«h¥²¥æ¤@ÂI ¦p2x+3y=5©M2x+4y=5 ¤°»ò®É­Ô««ª½©O? ·í¨â¤èµ{¦¡ªº±×²v­¼°_¨Ó¬°-1®É¡A¨â±ø½u´N·|««ª½Åo ®Ú¾Ú¤W¹Ï¥i¥Hµo²{¥kÃ䪺¤T¨¤§Î©M¥ªÃ䪺¤T¨¤§Î¬Û¦ü ¤S¥kÃä³o±ø¶Â½uªº±×²v¬Oa/b¥ªÃ䨺±ø¶Â½u¬O-d/c ¦ý¬O¨â¤T¨¤§Î¬Û¦ü¡A©Ò¥Ha/b=c/d¡A«h¨â±ø¶Â½uªº±×²v­¼°_¨Ó¬°-1 ¦p2x+3y=5©M3x-2y=5 ½Ð¦Û¤v¸Õ¸Õ§a ³o­Ó¤pµ{¦¡¬O§Q¥ÎGeoGebra°µ¥X¨Óªº¡]©xºô¡Gwww.geogebra.org¡^¡A¥¦»Ý­n¥Î¨ìJava¡A¦ý±z¨S¦³¦w¸ËJava¡A½Ð¦Ü¡Gwww.java.com¦w¸Ë¡C GeoGebra製作 ¤G¤¸¤@¦¸Áp¥ß¤èµ{¦¡¸Ñªº´X¦ó·N¸q ®Ú¾Ú¤W­±ªº¤¶²Ð¡Aª¾¹D¥i¥H§â¤G¤¸¤@¦¸¤èµ{¦¡µe¦b§¤¼Ð¥­­±¤W¬°¤@ª½½u ¨ºÁp¥ß¤èµ{¦¡ªº¸Ü¡A´N¥i¥H¦b§¤¼Ð¥­­±¤Wµe¥X¨â±ø½u ¤Sª¾¹D½u©Ò¥Nªíªº·N«ä¬O¤èµ{¦¡ªº¸Ñ ¦Ó¤G¤¸¤@¦¸¤èµ{¦¡ªº¸Ñ¬°º¡¨¬¨â­Ó¤èµ{¦¡ªº¸Ñ ¤]´N¬O³o¸Ñ¦pªGµe¦b§¤¼Ð¥­­±¤W·|¦P®É¦b¨â±ø½u¤W º¡¨¬³o±ø¥óªº´N¬O¥æÂI ¦]¦¹¤G¤¸¤@¦¸Áp¥ß¤èµ{¦¡¸Ñªº´X¦ó·N¸q´N¬O¨â±ø½uªº¥æÂI ¦p §â¥¦µe¦b§¤¼Ð¥­­±¤W ¦¹Áp¥ß¤èµ{¦¡ªº¸Ñ¬°x=25/13,y=5/13 §â¥¦µe¦b§¤¼Ð¥­­±¤W ´N·|µo²{¥¦¯uªº·|¬O¥æÂI ¦^­º­¶