{"id":585,"date":"2025-05-23T14:04:06","date_gmt":"2025-05-23T06:04:06","guid":{"rendered":"https:\/\/tilearn.space\/?p=585"},"modified":"2025-05-23T14:04:06","modified_gmt":"2025-05-23T06:04:06","slug":"%e6%9a%91%e5%81%87%e6%8f%90%e9%ab%981-%e7%ac%94%e8%ae%b0","status":"publish","type":"post","link":"https:\/\/tilearn.space\/?p=585","title":{"rendered":"\u6691\u5047\u63d0\u9ad81.\u7b14\u8bb0"},"content":{"rendered":"<style>\n  body {\n    font-family: sans-serif;\n    line-height: 1.6;\n    margin: 20px;\n  }\n  h2, h3, h4 {\n    color: #333;\n    margin-top: 20px;\n    margin-bottom: 10px;\n  }\n  ul {\n    margin-bottom: 15px;\n  }\n  li {\n    margin-bottom: 8px;\n  }\n  strong {\n    color: #555;\n  }\n  .math-formula {\n    margin: 10px 0;\n    overflow-x: auto; \/* \u9632\u6b62\u516c\u5f0f\u8fc7\u957f\u65f6\u6ea2\u51fa *\/\n  }\n<\/style>\n<h2>\u6bd4\u4f8b\u590d\u4e60<\/h2>\n<h3>\u4e00\u3001\u6bd4\u4f8b\u7ebf\u6bb5\u7684\u6027\u8d28<\/h3>\n<h4>1. \u57fa\u672c\u6027\u8d28\uff1a \\(\\frac{a}{b} = \\frac{c}{d} \\longrightarrow ad=bc\\)<\/h4>\n<ul>\n<li>\u6bd4\u4f8b\u5f0f\u4ea4\u53c9\u76f8\u4e58\uff0c\u6240\u5f97\u79ef\u76f8\u7b49\u3002\uff08\u8fd9\u662f\u6bd4\u4f8b\u6700\u57fa\u672c\u7684\u6027\u8d28\u548c\u5224\u65ad\u6bd4\u4f8b\u662f\u5426\u6210\u7acb\u7684\u4f9d\u636e\uff09<\/li>\n<li>\n<p>\u53cd\u4e4b\uff0c\u5982\u679c \\(ad=bc\\) (\u5176\u4e2d \\(b \\ne 0, d \\ne 0\\))\uff0c\u90a3\u4e48 \\(\\frac{a}{b}=\\frac{c}{d}\\) \u4e5f\u662f\u6210\u7acb\u7684\u3002<\/p>\n<p>\u7531\u6b64\u8fd8\u53ef\u4ee5\u63a8\u51fa\u5176\u4ed6\u6bd4\u4f8b\u5f0f\uff0c\u6bd4\u5982 \\(\\frac{a}{c}=\\frac{b}{d}\\) \u7b49\uff0c\u53ea\u8981\u4fdd\u8bc1\u4ea4\u53c9\u76f8\u4e58\u540e\u4ecd\u662f \\(ad=bc\\) \u5373\u53ef\u3002<\/p>\n<\/li>\n<\/ul>\n<h4>2. \u53cd\u6bd4\u6027\u8d28\uff1a \\(\\frac{a}{b} = \\frac{c}{d} \\longleftrightarrow \\frac{b}{a} = \\frac{d}{c}\\)<\/h4>\n<ul>\n<li>\u5982\u679c\u4e24\u4e2a\u6bd4\u76f8\u7b49\uff0c\u90a3\u4e48\u5b83\u4eec\u5404\u81ea\u7684\u5012\u6570\u4e5f\u76f8\u7b49\u3002<\/li>\n<li>\u4f7f\u7528\u53cd\u6bd4\u6027\u8d28\u540e\uff0c\u6bd4\u503c\u4f1a\u53d8\u4e3a\u539f\u6765\u7684\u5012\u6570\u3002\u9700\u8981\u6ce8\u610f \\(a, b, c, d\\) \u90fd\u4e0d\u80fd\u4e3a\u96f6\u3002<\/li>\n<\/ul>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u2235 \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u6839\u636e\u57fa\u672c\u6027\u8d28\uff0c\u5f97 \\(ad=bc\\)\u3002\u5c06\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5 \\(ac\\) (\u9700\u4fdd\u8bc1 \\(a \\ne 0, c \\ne 0\\))\uff0c\u5f97 \\(\\frac{ad}{ac} = \\frac{bc}{ac}\\)\uff0c\u5316\u7b80\u5f97 \\(\\frac{d}{c} = \\frac{b}{a}\\)\uff0c\u5373 \\(\\frac{b}{a} = \\frac{d}{c}\\)\u3002\u53cd\u4e4b\u540c\u7406\u53ef\u8bc1\u3002<\/p>\n<h4>*3. \u66f4\u6bd4\u6027\u8d28\uff1a \\(\\frac{a}{b} = \\frac{c}{d} \\longleftrightarrow \\frac{a}{c} = \\frac{b}{d}\\)<\/h4>\n<ul>\n<li>\u5982\u679c\u4e24\u4e2a\u6bd4\u76f8\u7b49\uff0c\u90a3\u4e48\u4ea4\u6362\u6bd4\u4f8b\u5185\u9879\u6216\u5916\u9879\u7684\u4f4d\u7f6e\uff0c\u6bd4\u4f8b\u4ecd\u7136\u6210\u7acb\u3002<\/li>\n<li>\u4f8b\u5982\uff0c\u5c06\u5185\u9879 b \u548c c \u4ea4\u6362\u4f4d\u7f6e\uff0c\u5f97\u5230 \\(\\frac{a}{c} = \\frac{b}{d}\\)\uff1b\u5c06\u5916\u9879 a \u548c d \u4ea4\u6362\u4f4d\u7f6e\uff0c\u5f97\u5230 \\(\\frac{d}{b} = \\frac{c}{a}\\)\u3002<\/li>\n<li>\u4f7f\u7528\u66f4\u6bd4\u6027\u8d28\u540e\uff0c\u6bd4\u503c\u4f1a\u53d1\u751f\u6539\u53d8\u3002<\/li>\n<\/ul>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u2235 \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u6839\u636e\u57fa\u672c\u6027\u8d28\uff0c\u5f97 \\(ad=bc\\)\u3002\u5c06\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5 \\(cd\\) (\u9700\u4fdd\u8bc1 \\(c \\ne 0, d \\ne 0\\))\uff0c\u5f97 \\(\\frac{ad}{cd} = \\frac{bc}{cd}\\)\uff0c\u5316\u7b80\u5f97 \\(\\frac{a}{c} = \\frac{b}{d}\\)\u3002\u53cd\u4e4b\u540c\u7406\u53ef\u8bc1\u3002<\/p>\n<h4>4. \u8bbek\u6cd5<\/h4>\n<p>\u8bbek\u6cd5\u662f\u5904\u7406\u6bd4\u4f8b\u95ee\u9898\u7684\u4e00\u79cd\u5e38\u7528\u6280\u5de7\uff0c\u6838\u5fc3\u601d\u60f3\u662f\u5f15\u5165\u53c2\u6570 k \u6765\u8868\u793a\u6bd4\u503c\uff0c\u4ece\u800c\u5c06\u6bd4\u4f8b\u5173\u7cfb\u8f6c\u5316\u4e3a\u7b49\u5f0f\u5173\u7cfb\uff0c\u65b9\u4fbf\u8fdb\u884c\u4ee3\u6570\u8fd0\u7b97\u3002<\/p>\n<ul>\n<li>\u5bf9\u4e8e\u4e24\u4e2a\u6bd4\u4f8b\u76f8\u7b49\u7684\u60c5\u51b5\uff0c\u5982 \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u53ef\u4ee5\u8bbe\u6bd4\u503c\u4e3a k\uff0c\u5373 \\(\\frac{a}{b} = \\frac{c}{d} = k\\) (\u5176\u4e2d \\(k \\ne 0\\))\u3002\u7531\u6b64\u53ef\u5f97 \\(a = bk\\)\uff0c\\(c = dk\\)\u3002<\/li>\n<li>\u5bf9\u4e8e\u591a\u4e2a\u6bd4\u503c\u76f8\u7b49\u7684\u60c5\u51b5\uff08\u7b49\u6bd4\uff09\uff0c\u5982 \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n}\\)\uff0c\u53ef\u4ee5\u8bbe\u516c\u6bd4\u4e3a k\uff0c\u5373 \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n} = k\\) (\u5176\u4e2d \\(k \\ne 0\\)\uff0c\u4e14 \\(b_i \\ne 0\\))\u3002\u7531\u6b64\u53ef\u5f97 \\(a_1 = b_1 k\\)\uff0c\\(a_2 = b_2 k\\)\uff0c&#8230;\uff0c\\(a_n = b_n k\\)\u3002<\/li>\n<li>\n<p>\u4f8b\u5982\uff1a\u82e5 \\(\\frac{a}{2} = \\frac{b}{3} = \\frac{c}{4}\\)\uff0c\u53ef\u8bbe \\(\\frac{a}{2} = \\frac{b}{3} = \\frac{c}{4} = k\\)\u3002<\/p>\n<p>\u5219 \\(a=2k\\)\uff0c\\(b=3k\\)\uff0c\\(c=4k\\)\u3002<\/p>\n<\/li>\n<\/ul>\n<h4>5. \u5408\u6bd4\u3001\u5206\u6bd4\u4e0e\u5408\u5206\u6bd4\u6027\u8d28<\/h4>\n<p>\u5982\u679c \\(\\frac{a}{b} = \\frac{c}{d}\\) (\u5176\u4e2d \\(b \\ne 0, d \\ne 0\\))\uff0c\u90a3\u4e48\uff1a<\/p>\n<ul>\n<li><strong>\u5408\u6bd4\u6027\u8d28\uff1a<\/strong> \\(\\frac{a+b}{b} = \\frac{c+d}{d}\\)<\/li>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u56e0\u4e3a \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u6240\u4ee5\u5728\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u52a0 1\uff0c\u5f97 \\(\\frac{a}{b} + 1 = \\frac{c}{d} + 1\\)\uff0c\u5373 \\(\\frac{a+b}{b} = \\frac{c+d}{d}\\)\u3002<\/p>\n<li><strong>\u5206\u6bd4\u6027\u8d28\uff1a<\/strong> \\(\\frac{a-b}{b} = \\frac{c-d}{d}\\)<\/li>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u56e0\u4e3a \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u6240\u4ee5\u5728\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u51cf 1\uff0c\u5f97 \\(\\frac{a}{b} &#8211; 1 = \\frac{c}{d} &#8211; 1\\)\uff0c\u5373 \\(\\frac{a-b}{b} = \\frac{c-d}{d}\\)\u3002<\/p>\n<li><strong>\u5408\u5206\u6bd4\u6027\u8d28\uff1a<\/strong> \u5982\u679c \\(a \\ne b\\) \u4e14 \\(c \\ne d\\)\uff0c\u90a3\u4e48 \\(\\frac{a+b}{a-b} = \\frac{c+d}{c-d}\\)<\/li>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u7531\u5408\u6bd4\u6027\u8d28\u5f97 \\(\\frac{a+b}{b} = \\frac{c+d}{d}\\)\uff0c\u7531\u5206\u6bd4\u6027\u8d28\u5f97 \\(\\frac{a-b}{b} = \\frac{c-d}{d}\\)\u3002\u5c06\u5408\u6bd4\u6027\u8d28\u7684\u7b49\u5f0f\u9664\u4ee5\u5206\u6bd4\u6027\u8d28\u7684\u7b49\u5f0f\uff08\u7b49\u53f7\u4e24\u8fb9\u7684\u975e\u96f6\u9879\u76f8\u9664\uff09\uff0c\u5373\u53ef\u5f97\u5230 \\(\\frac{\\frac{a+b}{b}}{\\frac{a-b}{b}} = \\frac{\\frac{c+d}{d}}{\\frac{c-d}{d}}\\)\uff0c\u5316\u7b80\u5f97 \\(\\frac{a+b}{a-b} = \\frac{c+d}{c-d}\\)\u3002\u6ce8\u610f\u5206\u6bcd \\(a-b \\ne 0\\) \u4e14 \\(c-d \\ne 0\\)\u3002<\/p>\n<\/ul>\n<h4>\u66f4\u4e00\u822c\u7684\u5f62\u5f0f (\u63a8\u5e7f)\uff1a<\/h4>\n<p>\u5982\u679c \\(\\frac{a}{b} = \\frac{c}{d}\\) (\u5176\u4e2d \\(b \\ne 0, d \\ne 0\\))\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4efb\u610f\u5e38\u6570 \\(x, y, m, n\\)\uff0c\u4e14 \\(m \\cdot a + n \\cdot b \\ne 0, m \\cdot c + n \\cdot d \\ne 0\\)\uff0c\u6709\uff1a<\/p>\n<p>\\[ \\frac{x \\cdot a + y \\cdot b}{m \\cdot a + n \\cdot b} = \\frac{x \\cdot c + y \\cdot d}{m \\cdot c + n \\cdot d} \\]<\/p>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u2235 \\(\\frac{a}{b} = \\frac{c}{d}\\)\uff0c\u7531\u8bbek\u6cd5\u53ef\u8bbe \\(\\frac{a}{b} = \\frac{c}{d} = k\\) (\u5176\u4e2d \\(k \\ne 0\\))\u3002\u5219 \\(a = bk\\)\uff0c\\(c = dk\\)\u3002<\/p>\n<p>\u5de6\u5f0f = \\(\\frac{x \\cdot a + y \\cdot b}{m \\cdot a + n \\cdot b} = \\frac{x \\cdot (bk) + y \\cdot b}{m \\cdot (bk) + n \\cdot b} = \\frac{b(xk+y)}{b(mk+n)} = \\frac{xk+y}{mk+n}\\)<\/p>\n<p>\u53f3\u5f0f = \\(\\frac{x \\cdot c + y \\cdot d}{m \\cdot c + n \\cdot d} = \\frac{x \\cdot (dk) + y \\cdot d}{m \\cdot (dk) + n \\cdot d} = \\frac{d(xk+y)}{d(mk+n)} = \\frac{xk+y}{mk+n}\\)<\/p>\n<p>\u2234 \u5de6\u5f0f = \u53f3\u5f0f\u3002\u6ce8\u610f\u5206\u6bcd \\(m \\cdot a + n \\cdot b \\ne 0\\) \u4e14 \\(m \\cdot c + n \\cdot d \\ne 0\\)\u3002<\/p>\n<p>\u4f7f\u7528\u5408\u6bd4\u3001\u5206\u6bd4\u3001\u5408\u5206\u6bd4\u6027\u8d28\u4ee5\u53ca\u5176\u63a8\u5e7f\u5f62\u5f0f\u540e\uff0c\u6bd4\u503c\u901a\u5e38\u4f1a\u53d1\u751f\u6539\u53d8\uff08\u7b49\u6bd4\u6027\u8d28\u9664\u5916\uff09\u3002<\/p>\n<p>\u5728\u8fdb\u884c\u6bd4\u4f8b\u53d8\u5f62\u65f6\uff0c\u4e00\u5b9a\u8981\u6ce8\u610f\u5206\u6bcd\u4e0d\u80fd\u4e3a\u96f6\u3002<\/p>\n<h4>6. \u7b49\u6bd4\u6027\u8d28<\/h4>\n<p>\u5982\u679c \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n}\\) (\u5176\u4e2d \\(b_i \\ne 0\\))\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4efb\u610f\u4e0d\u5168\u4e3a\u96f6\u7684\u5e38\u6570 \\(m_1, m_2, \\dots, m_n\\)\uff0c\u4e14 \\(m_1 b_1 + m_2 b_2 + \\dots + m_n b_n \\ne 0\\)\uff0c\u6709\uff1a<\/p>\n<p>\\[ \\frac{m_1 a_1 + m_2 a_2 + \\dots + m_n a_n}{m_1 b_1 + m_2 b_2 + \\dots + m_n b_n} = \\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n} \\]<\/p>\n<ul>\n<li>\u7b49\u6bd4\u6027\u8d28\u662f\u8bf4\uff0c\u5728\u591a\u4e2a\u6bd4\u76f8\u7b49\u7684\u6761\u4ef6\u4e0b\uff0c\u5c06\u8fd9\u4e9b\u6bd4\u7684\u524d\u9879\u548c\u540e\u9879\u5206\u522b\u6309\u4efb\u610f\u4e00\u7ec4\u4e0d\u5168\u4e3a\u96f6\u7684\u4e58\u6570\u76f8\u52a0\uff0c\u6240\u5f97\u7684\u65b0\u6bd4\u4ecd\u7136\u7b49\u4e8e\u539f\u6765\u7684\u6bcf\u4e00\u4e2a\u6bd4\u3002<\/li>\n<li>\u4f7f\u7528\u7b49\u6bd4\u6027\u8d28\u540e\uff0c\u6bd4\u503c\u4e0d\u4f1a\u53d1\u751f\u6539\u53d8\u3002<\/li>\n<li>\u53d8\u5f62\u8fc7\u7a0b\u4e2d\uff0c\u5206\u6bcd\u5fc5\u987b\u4e0d\u4e3a\u96f6\u3002<\/li>\n<\/ul>\n<p><strong>\u7406\u8bba\u4f9d\u636e\uff1a<\/strong> \u2235 \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n}\\)\uff0c\u7531\u8bbek\u6cd5\u53ef\u8bbe \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n} = k\\) (\u5176\u4e2d \\(k \\ne 0\\))\u3002<\/p>\n<p>\u5219 \\(a_1 = b_1 k\\)\uff0c\\(a_2 = b_2 k\\)\uff0c&#8230;\uff0c\\(a_n = b_n k\\)\u3002<\/p>\n<p>\u2234 \\(\\frac{m_1 a_1 + m_2 a_2 + \\dots + m_n a_n}{m_1 b_1 + m_2 b_2 + \\dots + m_n b_n} = \\frac{m_1 (b_1 k) + m_2 (b_2 k) + \\dots + m_n (b_n k)}{m_1 b_1 + m_2 b_2 + \\dots + m_n b_n} = \\frac{k(m_1 b_1 + m_2 b_2 + \\dots + m_n b_n)}{m_1 b_1 + m_2 b_2 + \\dots + m_n b_n}\\)<\/p>\n<p>\u5728\u5206\u6bcd \\(m_1 b_1 + m_2 b_2 + \\dots + m_n b_n \\ne 0\\) \u7684\u60c5\u51b5\u4e0b\uff0c\u4e0a\u5f0f\u5316\u7b80\u5f97 \\(k\\)\u3002<\/p>\n<p>\u800c \\(k = \\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n}\\)\u3002<\/p>\n<p>\u6240\u4ee5 \\(\\frac{m_1 a_1 + m_2 a_2 + \\dots + m_n a_n}{m_1 b_1 + m_2 b_2 + \\dots + m_n b_n} = \\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\dots = \\frac{a_n}{b_n}\\)\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6bd4\u4f8b\u590d\u4e60 \u4e00\u3001\u6bd4\u4f8b\u7ebf\u6bb5\u7684\u6027\u8d28 1. \u57fa\u672c\u6027\u8d28\uff1a \\(\\frac{a}{b} = \\frac{c}{d} \\lo [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[33,32,28],"tags":[],"class_list":["post-585","post","type-post","status-publish","format-standard","hentry","category-zkcc","category-jnj","category-fnjzl"],"_links":{"self":[{"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/posts\/585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tilearn.space\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=585"}],"version-history":[{"count":1,"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions"}],"predecessor-version":[{"id":586,"href":"https:\/\/tilearn.space\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions\/586"}],"wp:attachment":[{"href":"https:\/\/tilearn.space\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tilearn.space\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tilearn.space\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}