2192:
788:
195:
663:
1156:
1082:
858:
333:
440:
1488:
116:
1851:
947:
714:
2064:
1386:
times the second equation equals the third equation. Since this always holds for the right sides, all of which are 0, we merely need to require it to hold for the left sides as well:
1940:
1209:
553:
992:
1361:
1316:
504:
1679:
1593:
1268:
2045:
1991:
1632:
1548:
584:
904:
1716:
times the original second equation equals the original third equation; we conclude that the third equation is linearly dependent on the first two.
716:
to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of
730:
127:
1743:
satisfying all three equations in the parameters, and therefore the third original equation would be independent of the first two.
595:
1090:
1016:
673:
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the
799:
230:
2221:
348:
1392:
1727:) = (1, 1) that satisfied the first two equations in the parameters would not have satisfied the third one (
2187:{\displaystyle e={\frac {ac+bd}{c^{2}+d^{2}}}\quad \quad {\text{and}}\quad \quad f={\frac {bc-ad}{c^{2}+d^{2}}}.}
58:
1789:
909:
679:
49:
1501:
on both sides, and equating the constants on both sides gives the following system in the desired parameters
1862:
1719:
Note that if the constant term in the original third equation had been anything other than –7, the values (
1167:
2240:
458:
2213:
445:
At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0
510:
952:
1322:
1274:
467:
1643:
1554:
1229:
2009:
1955:
1599:
1700:) = (1, 1); since these values also satisfy the third equation, there do in fact exist
1515:
1220:
559:
2205:
874:
995:
8:
2206:
1007:
32:. It relies on the fact that two expressions are identical precisely when corresponding
2245:
2217:
1367:
717:
1997:
1752:
1162:
674:
2234:
2058:, and they can be solved to obtain the desired coefficients of the quotient:
1946:
33:
17:
1214:
36:
are equal for each different type of term. The method is used to bring
25:
24:
is a way of solving a functional equation of two expressions such as
29:
1751:
The method of equating coefficients is often used when dealing with
453: + 1, having zero coefficients for the positive powers of
224: − 2) turns both into polynomials, which we equate:
783:{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}={\sqrt {d}}+{\sqrt {e}}.}
190:{\displaystyle {\frac {A}{x}}+{\frac {B}{x-1}}+{\frac {C}{x-2}},\,}
37:
457:. Equating the corresponding coefficients now results in this
338:
or, after expansion and collecting terms with equal powers of
658:{\displaystyle A={\frac {1}{2}},\,B=-1,\,C={\frac {1}{2}}.\,}
1771:, we postulate that the ratio equals the complex number
1151:{\displaystyle d={\frac {a-{\sqrt {a^{2}-b^{2}c}}}{2}},}
1077:{\displaystyle e={\frac {a+{\sqrt {a^{2}-b^{2}c}}}{2}},}
2067:
2012:
1958:
1865:
1856:
and multiply both sides by the denominator to obtain
1792:
1646:
1602:
1557:
1518:
1395:
1325:
1277:
1232:
1215:
Example of testing for linear dependence of equations
1170:
1093:
1019:
955:
912:
877:
802:
733:
682:
598:
562:
513:
470:
351:
233:
130:
61:
1775:, and we wish to find the values of the parameters
871:we equate the terms not involving square roots, so
2186:
2050:These are two equations in the unknown parameters
2039:
1985:
1934:
1845:
1673:
1626:
1587:
1542:
1482:
1355:
1310:
1262:
1203:
1150:
1076:
986:
941:
898:
852:
782:
708:
657:
578:
547:
498:
434:
327:
189:
110:
853:{\displaystyle a+b{\sqrt {c}}=d+e+2{\sqrt {de}}.}
2232:
328:{\displaystyle A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,\,}
435:{\displaystyle (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,}
1746:
1483:{\displaystyle a(x-2y+1)+b(3x+5y-8)=4x+3y-7.}
793:Squaring both sides of this equation yields:
668:
1755:. For example, to divide the complex number
1497:on both sides, equating the coefficients of
906:and equate the parts involving radicals, so
121:that is, we want to bring it into the form:
43:
1370:on the first two, postulate two parameters
111:{\displaystyle {\frac {1}{x(x-1)(x-2)}},\,}
1846:{\displaystyle {\frac {a+bi}{c+di}}=e+fi,}
998:and one linear, in the desired parameters
942:{\displaystyle b{\sqrt {c}}=2{\sqrt {de}}}
709:{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}}
654:
634:
618:
575:
544:
495:
431:
324:
186:
107:
1712:times the original first equation plus
1692:satisfying the first two equations is (
1223:(with 3 equations in just 2 unknowns):
2233:
2203:
1935:{\displaystyle (ce-fd)+(ed+cf)i=a+bi.}
1366:To test whether the third equation is
1204:{\displaystyle {\sqrt {a^{2}-b^{2}c}}}
1735:= constant), so there would exist no
200:in which the unknown parameters are
13:
1221:overdetermined system of equations
14:
2257:
1996:and equating coefficients of the
1783:for which this is true. We write
994:This gives us two equations, one
212:. Multiplying these formulas by
2128:
2127:
2121:
2120:
1161:which is a valid solution pair
1908:
1890:
1884:
1866:
1453:
1429:
1420:
1399:
1382:times the first equation plus
410:
386:
370:
352:
312:
300:
288:
276:
264:
252:
249:
237:
98:
86:
83:
71:
50:partial fraction decomposition
1:
2197:
1493:Equating the coefficients of
548:{\displaystyle 3A+2B+C=0,\,}
7:
2208:Encyclopedia of Mathematics
987:{\displaystyle b^{2}c=4de.}
949:which when squared implies
10:
2262:
1747:Example in complex numbers
1684:The unique pair of values
1356:{\displaystyle 4x+3y-7=0.}
1311:{\displaystyle 3x+5y-8=0,}
669:Example in nested radicals
499:{\displaystyle A+B+C=0,\,}
459:system of linear equations
2212:. Facts on File. p.
1674:{\displaystyle a=1,\ b=1}
1588:{\displaystyle -2a+5b=3,}
1263:{\displaystyle x-2y+1=0,}
48:Suppose we want to apply
44:Example in real fractions
22:equating the coefficients
2040:{\displaystyle ed+cf=b.}
1986:{\displaystyle ce-fd=a,}
1627:{\displaystyle a-8b=-7.}
28:for a number of unknown
1543:{\displaystyle a+3b=4,}
589:Solving it results in:
579:{\displaystyle 2A=1.\,}
2204:Tanton, James (2005).
2188:
2041:
1987:
1936:
1847:
1763:by the complex number
1675:
1628:
1589:
1544:
1484:
1357:
1312:
1264:
1211:is a rational number.
1205:
1152:
1078:
988:
943:
900:
899:{\displaystyle a=d+e,}
854:
784:
710:
659:
580:
549:
500:
436:
329:
220: − 1)(
191:
112:
2189:
2042:
1988:
1937:
1848:
1676:
1629:
1590:
1545:
1485:
1358:
1313:
1265:
1206:
1153:
1079:
989:
944:
901:
855:
785:
711:
660:
581:
550:
501:
437:
330:
192:
113:
40:into a desired form.
2065:
2010:
1956:
1863:
1790:
1644:
1600:
1555:
1516:
1393:
1323:
1275:
1230:
1168:
1091:
1017:
953:
910:
875:
800:
731:
680:
596:
560:
511:
468:
349:
231:
128:
59:
52:to the expression:
2241:Elementary algebra
2184:
2037:
1983:
1932:
1843:
1671:
1637:Solving it gives:
1624:
1585:
1540:
1480:
1368:linearly dependent
1353:
1308:
1260:
1201:
1148:
1074:
984:
939:
896:
850:
780:
706:
655:
576:
545:
496:
432:
325:
187:
108:
2179:
2125:
2118:
1823:
1661:
1199:
1143:
1137:
1069:
1063:
937:
921:
845:
817:
775:
765:
755:
754:
750:
704:
703:
699:
649:
613:
181:
160:
139:
102:
2253:
2227:
2211:
2193:
2191:
2190:
2185:
2180:
2178:
2177:
2176:
2164:
2163:
2153:
2136:
2126:
2123:
2119:
2117:
2116:
2115:
2103:
2102:
2092:
2075:
2046:
2044:
2043:
2038:
1992:
1990:
1989:
1984:
1941:
1939:
1938:
1933:
1852:
1850:
1849:
1844:
1824:
1822:
1808:
1794:
1680:
1678:
1677:
1672:
1659:
1633:
1631:
1630:
1625:
1594:
1592:
1591:
1586:
1549:
1547:
1546:
1541:
1489:
1487:
1486:
1481:
1362:
1360:
1359:
1354:
1317:
1315:
1314:
1309:
1269:
1267:
1266:
1261:
1210:
1208:
1207:
1202:
1200:
1195:
1194:
1182:
1181:
1172:
1157:
1155:
1154:
1149:
1144:
1139:
1138:
1133:
1132:
1120:
1119:
1110:
1101:
1083:
1081:
1080:
1075:
1070:
1065:
1064:
1059:
1058:
1046:
1045:
1036:
1027:
993:
991:
990:
985:
965:
964:
948:
946:
945:
940:
938:
930:
922:
917:
905:
903:
902:
897:
859:
857:
856:
851:
846:
838:
818:
813:
789:
787:
786:
781:
776:
771:
766:
761:
756:
752:
751:
746:
735:
715:
713:
712:
707:
705:
701:
700:
695:
684:
664:
662:
661:
656:
650:
642:
614:
606:
585:
583:
582:
577:
554:
552:
551:
546:
505:
503:
502:
497:
441:
439:
438:
433:
382:
381:
334:
332:
331:
326:
196:
194:
193:
188:
182:
180:
166:
161:
159:
145:
140:
132:
117:
115:
114:
109:
103:
101:
63:
20:, the method of
2261:
2260:
2256:
2255:
2254:
2252:
2251:
2250:
2231:
2230:
2224:
2200:
2172:
2168:
2159:
2155:
2154:
2137:
2135:
2122:
2111:
2107:
2098:
2094:
2093:
2076:
2074:
2066:
2063:
2062:
2011:
2008:
2007:
1957:
1954:
1953:
1864:
1861:
1860:
1809:
1795:
1793:
1791:
1788:
1787:
1753:complex numbers
1749:
1645:
1642:
1641:
1601:
1598:
1597:
1556:
1553:
1552:
1517:
1514:
1513:
1394:
1391:
1390:
1324:
1321:
1320:
1276:
1273:
1272:
1231:
1228:
1227:
1217:
1190:
1186:
1177:
1173:
1171:
1169:
1166:
1165:
1128:
1124:
1115:
1111:
1109:
1102:
1100:
1092:
1089:
1088:
1054:
1050:
1041:
1037:
1035:
1028:
1026:
1018:
1015:
1014:
960:
956:
954:
951:
950:
929:
916:
911:
908:
907:
876:
873:
872:
837:
812:
801:
798:
797:
770:
760:
745:
734:
732:
729:
728:
694:
683:
681:
678:
677:
675:nested radicals
671:
641:
605:
597:
594:
593:
561:
558:
557:
512:
509:
508:
469:
466:
465:
377:
373:
350:
347:
346:
232:
229:
228:
170:
165:
149:
144:
131:
129:
126:
125:
67:
62:
60:
57:
56:
46:
12:
11:
5:
2259:
2249:
2248:
2243:
2229:
2228:
2222:
2199:
2196:
2195:
2194:
2183:
2175:
2171:
2167:
2162:
2158:
2152:
2149:
2146:
2143:
2140:
2134:
2131:
2114:
2110:
2106:
2101:
2097:
2091:
2088:
2085:
2082:
2079:
2073:
2070:
2048:
2047:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
1998:imaginary unit
1994:
1993:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1943:
1942:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1854:
1853:
1842:
1839:
1836:
1833:
1830:
1827:
1821:
1818:
1815:
1812:
1807:
1804:
1801:
1798:
1748:
1745:
1682:
1681:
1670:
1667:
1664:
1658:
1655:
1652:
1649:
1635:
1634:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1595:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1550:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1491:
1490:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1364:
1363:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1318:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1270:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1219:Consider this
1216:
1213:
1198:
1193:
1189:
1185:
1180:
1176:
1163:if and only if
1159:
1158:
1147:
1142:
1136:
1131:
1127:
1123:
1118:
1114:
1108:
1105:
1099:
1096:
1085:
1084:
1073:
1068:
1062:
1057:
1053:
1049:
1044:
1040:
1034:
1031:
1025:
1022:
983:
980:
977:
974:
971:
968:
963:
959:
936:
933:
928:
925:
920:
915:
895:
892:
889:
886:
883:
880:
861:
860:
849:
844:
841:
836:
833:
830:
827:
824:
821:
816:
811:
808:
805:
791:
790:
779:
774:
769:
764:
759:
749:
744:
741:
738:
698:
693:
690:
687:
670:
667:
666:
665:
653:
648:
645:
640:
637:
633:
630:
627:
624:
621:
617:
612:
609:
604:
601:
587:
586:
574:
571:
568:
565:
555:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
506:
494:
491:
488:
485:
482:
479:
476:
473:
449: + 0
443:
442:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
380:
376:
372:
369:
366:
363:
360:
357:
354:
336:
335:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
198:
197:
185:
179:
176:
173:
169:
164:
158:
155:
152:
148:
143:
138:
135:
119:
118:
106:
100:
97:
94:
91:
88:
85:
82:
79:
76:
73:
70:
66:
45:
42:
9:
6:
4:
3:
2:
2258:
2247:
2244:
2242:
2239:
2238:
2236:
2225:
2223:0-8160-5124-0
2219:
2215:
2210:
2209:
2202:
2201:
2181:
2173:
2169:
2165:
2160:
2156:
2150:
2147:
2144:
2141:
2138:
2132:
2129:
2112:
2108:
2104:
2099:
2095:
2089:
2086:
2083:
2080:
2077:
2071:
2068:
2061:
2060:
2059:
2057:
2053:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2006:
2005:
2004:
2002:
1999:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1952:
1951:
1950:
1948:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1905:
1902:
1899:
1896:
1893:
1887:
1881:
1878:
1875:
1872:
1869:
1859:
1858:
1857:
1840:
1837:
1834:
1831:
1828:
1825:
1819:
1816:
1813:
1810:
1805:
1802:
1799:
1796:
1786:
1785:
1784:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1744:
1742:
1738:
1734:
1730:
1726:
1722:
1717:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1668:
1665:
1662:
1656:
1653:
1650:
1647:
1640:
1639:
1638:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1596:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1551:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1512:
1511:
1510:
1508:
1504:
1500:
1496:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1426:
1423:
1417:
1414:
1411:
1408:
1405:
1402:
1396:
1389:
1388:
1387:
1385:
1381:
1377:
1373:
1369:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1319:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1271:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1226:
1225:
1224:
1222:
1212:
1196:
1191:
1187:
1183:
1178:
1174:
1164:
1145:
1140:
1134:
1129:
1125:
1121:
1116:
1112:
1106:
1103:
1097:
1094:
1087:
1086:
1071:
1066:
1060:
1055:
1051:
1047:
1042:
1038:
1032:
1029:
1023:
1020:
1013:
1012:
1011:
1009:
1008:can be solved
1005:
1001:
997:
981:
978:
975:
972:
969:
966:
961:
957:
934:
931:
926:
923:
918:
913:
893:
890:
887:
884:
881:
878:
870:
866:
847:
842:
839:
834:
831:
828:
825:
822:
819:
814:
809:
806:
803:
796:
795:
794:
777:
772:
767:
762:
757:
747:
742:
739:
736:
727:
726:
725:
723:
719:
696:
691:
688:
685:
676:
651:
646:
643:
638:
635:
631:
628:
625:
622:
619:
615:
610:
607:
602:
599:
592:
591:
590:
572:
569:
566:
563:
556:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
507:
492:
489:
486:
483:
480:
477:
474:
471:
464:
463:
462:
460:
456:
452:
448:
428:
425:
422:
419:
416:
413:
407:
404:
401:
398:
395:
392:
389:
383:
378:
374:
367:
364:
361:
358:
355:
345:
344:
343:
341:
321:
318:
315:
309:
306:
303:
297:
294:
291:
285:
282:
279:
273:
270:
267:
261:
258:
255:
246:
243:
240:
234:
227:
226:
225:
223:
219:
215:
211:
207:
203:
183:
177:
174:
171:
167:
162:
156:
153:
150:
146:
141:
136:
133:
124:
123:
122:
104:
95:
92:
89:
80:
77:
74:
68:
64:
55:
54:
53:
51:
41:
39:
35:
31:
27:
23:
19:
2207:
2055:
2051:
2049:
2000:
1995:
1949:terms gives
1944:
1855:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1750:
1740:
1736:
1732:
1728:
1724:
1720:
1718:
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1683:
1636:
1506:
1502:
1498:
1494:
1492:
1383:
1379:
1375:
1371:
1365:
1218:
1160:
1006:, and these
1003:
999:
868:
864:
862:
792:
721:
672:
588:
454:
450:
446:
444:
339:
337:
221:
217:
213:
209:
205:
201:
199:
120:
47:
34:coefficients
21:
15:
720:parameters
26:polynomials
18:mathematics
2235:Categories
2198:References
1708:such that
1378:such that
1010:to obtain
724:such that
30:parameters
2246:Equations
2145:−
1966:−
1945:Equating
1876:−
1619:−
1607:−
1559:−
1475:−
1448:−
1406:−
1342:−
1294:−
1237:−
1184:−
1122:−
1107:−
1048:−
996:quadratic
626:−
384:−
307:−
283:−
259:−
244:−
175:−
154:−
93:−
78:−
863:To find
718:rational
38:formulas
2220:
2003:gives
1660:
753:
702:
2218:ISBN
2054:and
1947:real
1779:and
1773:e+fi
1374:and
1002:and
867:and
722:d, e
208:and
2214:162
2124:and
1731:– 8
16:In
2237::
2216:.
1769:di
1761:bi
1739:,
1723:,
1704:,
1696:,
1688:,
1622:7.
1509::
1505:,
1478:7.
1351:0.
573:1.
461::
429:1.
342::
204:,
2226:.
2182:.
2174:2
2170:d
2166:+
2161:2
2157:c
2151:d
2148:a
2142:c
2139:b
2133:=
2130:f
2113:2
2109:d
2105:+
2100:2
2096:c
2090:d
2087:b
2084:+
2081:c
2078:a
2072:=
2069:e
2056:f
2052:e
2035:.
2032:b
2029:=
2026:f
2023:c
2020:+
2017:d
2014:e
2001:i
1981:,
1978:a
1975:=
1972:d
1969:f
1963:e
1960:c
1930:.
1927:i
1924:b
1921:+
1918:a
1915:=
1912:i
1909:)
1906:f
1903:c
1900:+
1897:d
1894:e
1891:(
1888:+
1885:)
1882:d
1879:f
1873:e
1870:c
1867:(
1841:,
1838:i
1835:f
1832:+
1829:e
1826:=
1820:i
1817:d
1814:+
1811:c
1806:i
1803:b
1800:+
1797:a
1781:f
1777:e
1767:+
1765:c
1759:+
1757:a
1741:b
1737:a
1733:b
1729:a
1725:b
1721:a
1714:b
1710:a
1706:b
1702:a
1698:b
1694:a
1690:b
1686:a
1669:1
1666:=
1663:b
1657:,
1654:1
1651:=
1648:a
1616:=
1613:b
1610:8
1604:a
1583:,
1580:3
1577:=
1574:b
1571:5
1568:+
1565:a
1562:2
1538:,
1535:4
1532:=
1529:b
1526:3
1523:+
1520:a
1507:b
1503:a
1499:y
1495:x
1472:y
1469:3
1466:+
1463:x
1460:4
1457:=
1454:)
1451:8
1445:y
1442:5
1439:+
1436:x
1433:3
1430:(
1427:b
1424:+
1421:)
1418:1
1415:+
1412:y
1409:2
1403:x
1400:(
1397:a
1384:b
1380:a
1376:b
1372:a
1348:=
1345:7
1339:y
1336:3
1333:+
1330:x
1327:4
1306:,
1303:0
1300:=
1297:8
1291:y
1288:5
1285:+
1282:x
1279:3
1258:,
1255:0
1252:=
1249:1
1246:+
1243:y
1240:2
1234:x
1197:c
1192:2
1188:b
1179:2
1175:a
1146:,
1141:2
1135:c
1130:2
1126:b
1117:2
1113:a
1104:a
1098:=
1095:d
1072:,
1067:2
1061:c
1056:2
1052:b
1043:2
1039:a
1033:+
1030:a
1024:=
1021:e
1004:e
1000:d
982:.
979:e
976:d
973:4
970:=
967:c
962:2
958:b
935:e
932:d
927:2
924:=
919:c
914:b
894:,
891:e
888:+
885:d
882:=
879:a
869:e
865:d
848:.
843:e
840:d
835:2
832:+
829:e
826:+
823:d
820:=
815:c
810:b
807:+
804:a
778:.
773:e
768:+
763:d
758:=
748:c
743:b
740:+
737:a
697:c
692:b
689:+
686:a
652:.
647:2
644:1
639:=
636:C
632:,
629:1
623:=
620:B
616:,
611:2
608:1
603:=
600:A
570:=
567:A
564:2
542:,
539:0
536:=
533:C
530:+
527:B
524:2
521:+
518:A
515:3
493:,
490:0
487:=
484:C
481:+
478:B
475:+
472:A
455:x
451:x
447:x
426:=
423:A
420:2
417:+
414:x
411:)
408:C
405:+
402:B
399:2
396:+
393:A
390:3
387:(
379:2
375:x
371:)
368:C
365:+
362:B
359:+
356:A
353:(
340:x
322:,
319:1
316:=
313:)
310:1
304:x
301:(
298:x
295:C
292:+
289:)
286:2
280:x
277:(
274:x
271:B
268:+
265:)
262:2
256:x
253:(
250:)
247:1
241:x
238:(
235:A
222:x
218:x
216:(
214:x
210:C
206:B
202:A
184:,
178:2
172:x
168:C
163:+
157:1
151:x
147:B
142:+
137:x
134:A
105:,
99:)
96:2
90:x
87:(
84:)
81:1
75:x
72:(
69:x
65:1
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.