在我国古代数学著作《九章算术》有这样一个问题：上等谷3束中等谷2束下等谷1束共39斗。 求古代数学问题。
\u5728\u6211\u56fd\u53e4\u4ee3\u6570\u5b66\u8457\u4f5c\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\u6709\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898\uff1a\u4e0a\u7b49\u8c373\u675f\u4e2d\u7b49\u8c372\u675f\u4e0b\u7b49\u8c371\u675f\u517139\u6597\u4e0a\u7b49\u8c37\u4e24\u675f\u4e2d\u7b49\u8c373\u675f3x+2y+z=39
2x+3y+z=34
x+3z=26
1\u5f0f-2\u5f0f\u5f97x-y=5\uff08a\uff09
2\u5f0f\u00d73-3\u5f0f\u5f975x+9y=8\uff08b\uff09
a\u00d79+b=14x=13
\u63a5\u4e0b\u6765\u5e94\u8be5\u4f1a\u4e86\u5427
1.\u6211\u95ee\u5f00\u5e97\u674e\u516c\u516c\uff0c\u4f17\u5ba2\u6765\u5230\u5ba2\u5e97\u4e2d\u3002
\u4e00\u623f\u4e03\u5ba2\u591a\u4e03\u5ba2\uff0c\u4e00\u623f\u4e5d\u5ba2\u4e00\u623f\u7a7a\u3002
\u51e0\u95f4\u623f\u6765\u51e0\u4f4d\u5ba2\uff0c\u82e5\u541b\u77e5\u6653\u6211\u505a\u4e1c\u3002
2.\u4eca\u6709\u4e00\u7269\uff0c\u4e09\u4e09\u6570\u4f59\u4e8c\uff0c\u4e94\u4e94\u6570\u4f59\u4e09\uff0c\u4e03\u4e03\u6570\u4f59\u4e8c\uff0c\u95ee\u7269\u6709\u51e0\uff1f
--------------------------------------------------------------------------------
[\u6251(4)]\uff1a
\u5b59\u5b50\u7b97\u7ecf\uff1f\u5269\u4f59\u5b9a\u7406\uff1f\u697c\u4e3b\u5728\u505a\u5965\u6570\uff1f\u8fd8\u662f\u5728\u770b\u5c04\u96d5\uff1f\u8fd9\u9898\u76ee\u4e0d\u7528\u52b3\u70e6\u4e13\u4e1a\u4eba\u58eb\u4e86\uff0c\u9ad8\u4e2d\u5c31\u5e94\u8be5\u6559\u8fc7\u4e86\u5427\uff1f\u2026\u2026
\u539f\u6587\u89e3\u6cd5\uff1a\u672f\u66f0\uff1a\u300c\u4e09\u4e09\u6570\u4e4b\u5269\u4e8c\uff0c\u7f6e\u4e00\u767e\u56db\u5341\uff0c\u4e94\u4e94\u6570\u4e4b\u5269\u4e09\uff0c\u7f6e\u516d\u5341\u4e09\uff0c\u4e03\u4e03\u6570\u4e4b\u5269\u4e8c\uff0c\u7f6e\u4e09\u5341\uff0c\u5e76\u4e4b\uff0c\u5f97\u4e8c\u767e\u4e09\u5341\u4e09\uff0c\u4ee5\u4e8c\u767e\u4e00\u5341\u51cf\u4e4b\uff0c\u5373\u5f97\u3002\u51e1\u4e09\u4e09\u6570\u4e4b\u5269\u4e00\uff0c\u5219\u7f6e\u4e03\u5341\uff0c\u4e94\u4e94\u6570\u4e4b\u5269\u4e00\uff0c\u5219\u7f6e\u4e8c\u5341\u4e00\uff0c\u4e03\u4e03\u6570\u4e4b\u5269\u4e00\uff0c\u5219\u7f6e\u5341\u4e94\uff0c\u5373\u5f97\u3002\u300d
\u641e\u5b9a\u6536\u5de5\uff01
--------------------------------------------------------------------------------
[\u732b(5)]\uff1a
\u597d\u4eba\u505a\u5230\u5e95\uff0c\u7ffb\u8bd1\u4e00\u4e0b\u5427\u2026\u2026\u4ec0\u4e48\u610f\u601d\u5462\uff1f\u5c31\u662f\u8bf4\uff1a
\u5148\u505a\u4e00\u4e2a5\u548c7\u7684\u516c\u500d\u6570\uff0c\u4e14\u8981\u96643\u4f591\u7684\uff0c\u5f97\u523070\uff1b
\u7136\u540e\u505a\u4e00\u4e2a3\u548c7\u7684\u516c\u500d\u6570\uff0c\u4e14\u8981\u96645\u4f591\u7684\uff0c\u5f97\u523021\uff1b
\u6700\u540e\u505a\u4e00\u4e2a3\u548c5\u7684\u516c\u500d\u6570\uff0c\u4e14\u8981\u96647\u4f591\u7684\uff0c\u5f97\u523015\uff1b
\u7136\u540e\u6309\u9898\u76ee\u4e2d\u4f59\u6570\u7684\u5927\u5c0f\u5c06\u4e0a\u9762\u7684\u6570\u5b57\u500d\u5927\u518d\u76f8\u52a0\uff1a70*2+21*3+15*2=233
233\u5176\u5b9e\u5df2\u7ecf\u6ee1\u8db3\u6761\u4ef6\u4e86\uff0c\u4f46\u662f\u4e00\u822c\u6211\u4eec\u662f\u8981\u6700\u5c0f\u7684\uff0c\u600e\u4e48\u529e\u5462\uff1f\u5f88\u7b80\u5355\uff0c\u51cf3\u30015\u30017\u7684\u6700\u5c0f\u516c\u500d\u6570105\u76f4\u5230\u5f97\u51fa\u6700\u5c0f\u6574\u6570\u4e3a\u6b62\uff1a233-105*2=23
\u8fd9\u4e0b\u771f\u7684\u641e\u5b9a\u6536\u5de5\u4e86\uff013.\u9898\u76ee\u5e94\u8be5\u662f\u8fd9\u6837\u7684\uff1a
\u6709100\u4e2a\u548c\u5c1a\u5206100\u53ea\u9992\u5934\uff0c\u6b63\u597d\u5206\u5b8c\u3002\u5982\u679c\u5927\u548c\u5c1a\u4e00\u4eba\u52063\u53ea\uff0c\u5c0f\u548c\u5c1a3\u4eba\u5206\u4e00\u53ea\uff0c\u8bd5\u95ee\u5927\u3001\u5c0f\u548c\u5c1a\u5404\u6709\u51e0\u4eba\uff1f
--------------------------------------------------------------------------------\u672c\u95ee\u9898\u7684\u89e3\u6cd5\u751a\u591a\uff0c\u6700\u666e\u901a\u3001\u6700\u5e38\u89c4\u7684\u529e\u6cd5\u5f53\u7136\u662f\u5217\u51fa\u4e00\u4e2a\u65b9\u7a0b\u6765\u6c42\u89e3\uff0c\u8fd9\u5f88\u5bb9\u6613\u505a\u5230\uff0c\u4f46\u5176\u6d41\u5f0a\u662f\u4e00\u822c\u5316\u3001\u7a0b\u5f0f\u5316\uff0c\u5bf9\u5f00\u53d1\u667a\u529b\u4e0d\u5229\u3002
\u73b0\u5728\u4ecb\u7ecd\u4e00\u79cd\u522b\u5f00\u751f\u9762\u7684\u201c\u7f16\u7ec4\u6cd5\u201d\u3002\u300a\u76f4\u6307\u7b97\u6cd5\u7edf\u5b97\u300b\u91cc\u7684\u8bdd\u662f\uff1a\u201c\u7f6e\u50e7\u4e00\u767e\u4e3a\u5b9e\uff0c\u4ee5\u4e09\u4e00\u5e76\u5f97\u56db\u4e3a\u6cd5\u9664\u4e4b\uff0c\u5f97\u5927\u50e7\u4e8c\u5341\u4e94\u4e2a\u3002\u201d\u6240\u8c13\u201c\u5b9e\u201d\u4fbf\u662f\u201c\u88ab\u9664\u6570\u201d\uff0c\u201c\u6cd5\u201d\u4fbf\u662f\u201c\u9664\u6570\u201d\u3002\u5176\u529e\u6cd5\u662f\uff1a
100\u00f7\uff083+1\uff09=25\uff0c100-25=75\u3002
\u8fd9\u662f\u4e00\u79cd\u201c\u7f16\u7ec4\u6cd5\u201d\uff0c\u7531\u4e8e\u5927\u548c\u5c1a\u4e00\u4eba\u52063\u53ea\u9992\u5934\uff0c\u5c0f\u548c\u5c1a3\u4eba\u5206\u4e00\u53ea\u9992\u5934\u3002\u5408\u5e76\u8ba1\u7b97\uff0c\u5373\u662f\uff1a4\u4e2a\u548c\u5c1a\u54034\u53ea\u9992\u5934\u3002\u8fd9\u6837\uff0c100\u4e2a\u548c\u5c1a\u6b63\u597d\u7f16\u621025\u7ec4\uff0c\u800c\u6bcf\u4e00\u7ec4\u4e2d\u6070\u597d\u67091\u4e2a\u5927\u548c\u5c1a\uff0c\u6240\u4ee5\u4eba\u4eec\u7acb\u5373\u53ef\u7b97\u51fa\u5927\u548c\u5c1a\u670925\u4eba\uff0c\u4ece\u800c\u53ef\u77e5\u5c0f\u548c\u5c1a\u670975\u4eba\u30024.\u6709\u4e24\u4e2a\u4eba\u5408\u4f19\u5356\u9999\u6cb9\uff0c\u56e0\u95f9\u77db\u76fe\u51b3\u5b9a\u5206\u5f00\u3002\u5728\u6cb9\u7f38\u91cc\u670910\u65a4\u6cb9\uff0c\u8fd8\u6709\u4e00\u4e2a\u80fd\u88c57\u65a4\u6cb9\u7684\u6876\u548c\u4e00\u4e2a\u80fd\u88c53\u65a4\u6cb9\u7684\u58f6\uff0c\u6b64\u5916\u522b\u65e0\u5de5\u5177\uff0c\u8bf7\u95ee\u600e\u6837\u624d\u80fd\u5e73\u5747\u5206\u5f00\uff1f
\u5148\u5012\u6ee1\u4e03\u65a4\u7684\u6cb9\u6876\uff0c\u518d\u75283\u65a4\u7684\u6876\u8fde\u7eed\u5f8010\u65a4\u6876\u91cc\u5012\u4e24\u6b21\uff0c\u8fd9\u68377\u65a4\u6876\u91cc\u5c31\u52691\u65a4\uff0c\u7136\u540e\u628a\u8fd91\u65a4\u5012\u51653\u65a4\u6876\uff0c\u518d\u628a10\u65a4\u6876\u91cc\u9762\u76849\u65a4\u5012\u6ee17\u65a4\u6876\uff0c\u8fd9\u683710\u65a4\u6876\u5c31\u52692\u65a4\u4e86\uff0c\u7136\u540e\u75287\u65a4\u6876\u5012\u6ee13\u65a4\u540c\u6876\u540e\u5012\u516510\u65a4\u6876\u5c31\u51fa\u6765\u6b63\u597d\u4e24\u4e2a5\u65a4\u4e86
5.\u6807\u70b9\u7b26\u53f7\u662f\u4e66\u9762\u8bed\u91cc\u7528\u6765\u8868\u660e\u505c\u987f\u3001\u6807\u660e\u8bed\u6c14\u6216\u6807\u51fa\u8bcd\u8bed\u6027\u8d28\u4f5c\u7528\u7684\u7b26\u53f7\u3002\u6211\u4eec\u5728\u5199\u4f5c\u4e2d\u5c31\u7839\u6839\u636e\u6240\u8981\u8868\u8fbe\u7684\u610f\u4e49\u548c\u5177\u4f53\u7684\u8bed\u8a00\u73af\u5883\uff0c\u6070\u5207\u5730\u9009\u7528\u5408\u9002\u7684\u6807\u70b9\u3002
\u3000\u3000\u540c\u6837\uff0c\u6b63\u786e\u4f7f\u7528\u6807\u70b9\u7b26\u53f7\u5bf9\u89e3\u6570\u5b66\u9898\u4e5f\u6709\u5f88\u5927\u5e2e\u52a9\uff0c\u4e0b\u9762\u4e00\u9053\u53e4\u4ee3\u6570\u5b66\u9898\uff0c\u6ca1\u6709\u6807\u70b9\uff0c\u4f60\u80fd\u6b63\u786e\u6807\u51fa\u6807\u70b9\uff0c\u7136\u540e\u8ba1\u7b97\u51fa\u6765\u5417:
\u3000\u3000\u4e09\u89d2\u51e0\u4f55\u5171\u8ba1\u4e5d\u89d2\u4e09\u89d2\u4e09\u89d2\u51e0\u4f55\u51e0\u4f55
\u4e09\u89d2\uff0b\u51e0\u4f55\uff1d\u4e5d\u89d2\uff0c\u51e0\u4f55\uff1d\u4e5d\u89d2\uff0d\u4e09\u89d2\uff1d\u516d\u89d2
6.\u5728\u53e4\u5370\u5ea6\u6709\u4e00\u79cd\u8bd7\u6b4c\u5f62\u5f0f\u7684\u6570\u5b66\u9898\uff0c\u901a\u8fc7\u4eba\u4eec\u7684\u4f20\u9882\u5e7f\u6cdb\u6d41\u4f20\u7740\uff0e\u6709\u4e00\u4e2a\u9898\u76ee\u662f\u8fd9\u6837\u7684\uff1a
\u4e00\u7fa4\u7334\u5b50\u5206\u4e24\u961f\uff0c
\u9ad8\u9ad8\u5174\u5174\u505a\u6e38\u620f\uff0e
\u516b\u5206\u4e4b\u4e00\u518d\u5e73\u65b9\uff0c
\u8e66\u8e66\u8df3\u8df3\u8fdb\u6811\u6797\uff0e
\u5176\u4f59\u5341\u4e8c\u9ad8\u58f0\u558a\uff0c
\u5145\u6ee1\u6b22\u4e50\u7684\u6c14\u6c1b\u3002
\u544a\u6211\u603b\u6570\u662f\u591a\u5c11\uff0c
\u4e24\u961f\u7334\u5b50\u5728\u4e00\u8d77\uff0e
\u6211\u4eec\u8bbe\u7334\u5b50\u7684\u603b\u6570\u662fx\uff0c\u663e\u7136\u5168\u4f53\u7334\u5b50\u5206\u6210\u4e24\u4e2a\u90e8\u5206\uff0c\u4e0d\u96be\u5217\u51fa\u65b9\u7a0b
(x/8)2+12=x
\u89e3\u8fd9\u4e2a\u65b9\u7a0b\uff0c\u5f97x1=48\uff0cx2=16\uff0e\u7ecf\u68c0\u9a8c\uff0c\u8fd9\u4e24\u4e2a\u6839\u90fd\u7b26\u5408\u9898\u610f\uff0c\u6240\u4ee5\u7334\u5b50\u7684\u603b\u6570\u662f48\u621616\uff0e7.\u4e00\u5927\u4e00\u5c0f\u4e24\u53ea\u8001\u9f20\u6b63\u5bf9\u7740\u4e00\u5835\u5899\u6253\u6d1e,\u7b2c\u4e00\u5929,\u5927\u9f20\u548c\u5c0f\u9f20\u5404\u6253\u8fdb\u4e00\u5c3a,\u5927\u9f20\u8d8a\u5e72\u8d8a\u6709\u52b2,\u4ee5\u540e\u6bcf\u5929\u90fd\u589e\u52a0\u4e00\u500d,\u800c\u5c0f\u9f20\u5374\u8d8a\u5e72\u8d8a\u7d2f,\u6bcf\u5929\u7684\u8fdb\u5ea6\u90fd\u662f\u524d\u4e00\u5929\u7684\u4e00\u534a.\u5899\u539a\u4e94\u5c3a,\u95ee\u51e0\u5929\u4e4b\u540e,\u4e24\u53ea\u8001\u9f20\u80fd\u591f\u4f1a\u9762?\u5404\u6253\u8fdb\u51e0\u5c3a?
\u5934\u4e24\u5929\u5171\u6253\u6d1e4.5\u5c3a\uff0c
\u7b2c\u4e09\u5929\u5373
1+2+1+0.5+\uff084+1/4\uff09x=5
x\uff1d2/17\uff08\u96f6\u5934\u5929\u6570\uff09
\u5373\u9700\u89812.117647\u5929\u53732\u59292\u5c0f\u65f649\u520624.7\u79d28.\uff081\uff09\u4eca\u6709\u5171\u4e70\u9e21\uff0c\u8ba4\u51fa\u4e5d\uff0c\u76c8\u4e00\u5341\u4e00\uff1b\u4eba\u51fa\u516d\uff0c\u4e0d\u8db3\u5341\u516d\uff0c\u95ee\u4eba\u6570\u3001\u9e21\u4ef7\u5404\u51e0\u4f55\uff1f
\uff082\uff09\u4eca\u6709\u84b2\u751f\u4e00\u65e5\uff0c\u957f\u4e09\u5c3a\uff1b\u839e\u58f0\u4e00\u65e5\uff0c\u957f\u4e00\u5c3a\u3002\u84b2\u751f\u65e5\u81ea\u534a\uff0c\u839e\u751f\u65e5\u81ea\u500d\u3002\u95ee\u51e0\u4f55\u65e5\u800c\u957f\u7b49\uff1f 9.\u5e9e\u548c\u5b59\u90fd\u662f\u9b3c\u8c37\u5b50\u7684\u5f92\u5f1f\uff0c\u6709\u4e00\u5929\u4ed6\u51fa\u4e86\u4e00\u9053\u9898\u8ba9\u4ed6\u4eec\u505a\uff1a\u4ece2\u523099\u8fd9\u4e24\u4e2a\u6570\u7684\u8303\u56f4\u5185\u53d6\u4e24\u4e2a\u4e0d\u540c\u7684\u6570\u6c42\u4ed6\u4eec\u7684\u548c\u53ca\u79ef\uff0c\u9b3c\u8c37\u5b50\u53ea\u544a\u8bc9\u5e9e\u8fd9\u4e24\u4e2a\u6570\u7684\u548c\uff0c\u544a\u8bc9\u5b59\u8fd9\u4e24\u4e2a\u6570\u7684\u79ef\uff0c\u8ba9\u4ed6\u4eec\u6c42\u51fa\u8fd9\u4e24\u4e2a\u6570\u5206\u522b\u662f\u4ec0\u4e48\uff1f
\u4e24\u4e2a\u4eba\u89c1\u9762\u65f6\uff0c\u5e9e\u8bf4\uff1a\u201c\u6211\u867d\u7136\u4e0d\u77e5\u9053\u8fd9\u4e24\u4e2a\u6570\u662f\u4ec0\u4e48\uff0c\u5355\u6211\u80af\u5b9a\u4f60\u4e5f\u4e0d\u77e5\u9053\u3002\u201d
\u5b59\u8bf4\uff1a\u201c\u6ca1\u89c1\u5230\u4f60\u4e4b\u524d\u6211\u662f\u80af\u5b9a\u4e0d\u77e5\u9053\uff0c\u4f46\u73b0\u5728\u6211\u80af\u5b9a\u6211\u77e5\u9053\u4e86\u3002\u201d
\u5e9e\u8bf4\uff1a\u201c\u73b0\u5728\u770b\u5230\u4f60\uff0c\u6211\u4e5f\u80af\u5b9a\u77e5\u9053\u8fd9\u4e24\u4e2a\u6570\u662f\u4ec0\u4e48\u4e86\u3002\u201d
\u6c42\u8fd9\u4e24\u4e2a\u6570\u5206\u522b\u662f\u4ec0\u4e48\uff1f10.\u4e95\u7ef3\u4e00\u6839\uff0c\u4e09\u6298\u5165\u4e95\u5e95\u4f59\u4e00\u5c3a\uff0c\u56db\u6298\u5165\u4e95\u5e95\u5dee\u4e00\u5c3a\u3002\u7ef3\u957f\u51e0\u4f55\uff1b\u4e95\u6df1\u51e0\u4f55\uff1f\u6709\u610f\u541b\u7559\u89e3
解：设上等谷每束x斗，中等谷每束y斗，下等谷每束z斗。则
3x+2y+z=39 (1)
2x+3y+z=34 (2)
x+2y+3z=26 (3)
(1)-(2)，得 x-y =5 (4)
(1)乘3-(3)，得 8x+4y=91 (5)
(4)乘4+(5)，得 12x = 111，解得：x=9.25
x=9.25，代入(4)，解得 y=x-5 = 4.25
x=9.25，y=4.25，代入（1），解得 z=2.75
绛旓細璁炬按娣变负 x 灏猴紝鍒欒姦鑻囬暱 (x+1) 灏猴紝鐢卞嬀鑲″畾鐞嗘湁锛歺^2+5^2=(x+1)^2 瑙ｅ緱 x=12 鎵浠ユ按娣 12 灏猴紝鑺﹁媷闀 13 灏恒
绛旓細锛坆+1锛²-b²=5²=25 鍗2b+1=25锛岃В鍑篵=12锛堝昂锛夋墍浠ワ紝姘存睜娣12灏猴紝鑺﹁媷闀13灏
绛旓細鐒惰岋紝鍦ㄥ垬寰戒箣鍚庣殑璁稿鍙や唬鏁板钁椾綔涓紝鍒欎笉涔忔湁鏁板涓庛婂懆鏄撱嬪瘑鍒囩浉鍏崇殑浜嬩緥锛屽ぇ鑷村彲鍒嗕负浠ヤ笅涓夌被锛涓浗鍙や唬鏁板鑷眽浠銆婁節绔犵畻鏈褰㈡垚浣撶郴锛屽彂灞曡嚦瀹嬪厓鏃舵湡杈惧埌楂樺嘲锛屾湡闂村嚭鐜颁簡涓嶅皯閲嶈鐨勬暟瀛﹁憲浣溿傛眽鍞愭椂鏈熸湁鈥滅畻缁忓崄涔︹濓紝闄ゃ婁節绔犵畻鏈嬩箣澶栵紝杩樻湁銆婂懆楂绠楃粡銆嬨併婃捣宀涚畻缁忋嬨併婁簲鏇圭畻缁忋嬨併...
绛旓細璁捐姦鑻囬暱涓篨锛屾按姹犳繁涓簒-1 (10梅2)鐨勫钩鏂+X鐨勫钩鏂=(X+1)鐨勫钩鏂
绛旓細姘存睜涓ぎ璺濈姹犺竟涓5灏猴紝鑰岃姦鑻囬珮鍑烘按闈1灏猴紝璁捐姦鑻囬暱鏄疿锛屽垯锛圶-1)*2+5*2=X*2锛岀畻鍑烘潵寰楀埌X=13锛岄偅涔堟按娣卞氨鏄12锛岀湅鎴戠畻鐨勮繖涔堣緵鑻︼紝缁欏姞鍒嗗憲 璇烽噰绾崇瓟妗堬紝鏀寔鎴戜竴涓嬨
绛旓細寰堢畝鍗曪紝璁捐姦鑻囬暱x 鍕鹃瀹氱悊 鐩磋涓夎鍨嬬殑涓夎竟鍒嗗埆涓5 x-1 x (x-1)(x-1)+5*5=x*x x=13,姘存繁12
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