{"id":1158,"date":"2020-08-13T23:54:49","date_gmt":"2020-08-13T21:54:49","guid":{"rendered":"http:\/\/matma.com.pl\/?p=1158"},"modified":"2020-08-18T20:36:19","modified_gmt":"2020-08-18T18:36:19","slug":"wyrazenia-algebraiczne-rownania","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=1158","title":{"rendered":"Wyra\u017cenia algebraiczne. R\u00f3wnania"},"content":{"rendered":"<p><span style=\"font-size: 14pt;\"><strong><span style=\"font-family: 'book antiqua', palatino, serif; color: #4472c4;\">Wyra\u017cenia algebraiczne<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Witamy w algebrze!!!<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Od tego momentu b\u0119dzie to nieodzowna cz\u0119\u015b\u0107 matematyki, w kt\u00f3rej du\u017co &#8222;pracujemy&#8221; na literkach. Niejednokrotnie, w dotychczasowej nauce matematyki, korzystali\u015bcie z wyra\u017ce\u0144 algebraicznych. By\u0142y one dost\u0119pne m.in. na geometrii, w postaci wzor\u00f3w do obliczania pola czy obwodu.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Najprostszymi wyra\u017ceniami algebraicznymi s\u0105:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">litery (<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x,&amp;space;\\,&amp;space;y,\\,&amp;space;z\" alt=\"x, \\, y,\\, z\" align=\"absmiddle\" \/>),<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">liczby (<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?0,125,\\,\\,&amp;space;\\frac{3}{7},\\,5\" alt=\"0,125,\\,\\, \\frac{3}{7},\\,5\" align=\"absmiddle\" \/>) oraz<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"> po\u0142\u0105czone dzia\u0142aniami litery z liczbami (<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a+6,\\,&amp;space;\\frac{x}{7},&amp;space;\\,&amp;space;m^{2}\" alt=\"a+6,\\, \\frac{x}{7}, \\, m^{2}\" align=\"absmiddle\" \/>).<\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Jak stworzy\u0107 wyra\u017cenie algebraiczne? <\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Spr\u00f3bujmy zapisa\u0107 sum\u0119 dw\u00f3ch liczb <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a\" alt=\"a\" align=\"absmiddle\" \/> i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?b\" alt=\"b\" align=\"absmiddle\" \/>, mianowicie <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a+b\" alt=\"a+b\" align=\"absmiddle\" \/>. I w taki oto spos\u00f3b, uda\u0142o nam si\u0119 zapisa\u0107 pierwsze wyra\u017cenie algebraiczne. Spr\u00f3bujmy zapisa\u0107 wz\u00f3r na pole prostok\u0105ta o bokach d\u0142ugo\u015bci <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a\" alt=\"a\" align=\"absmiddle\" \/> i <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?b\" alt=\"b\" align=\"absmiddle\" \/>, mianowicie\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a\\cdot&amp;space;b\" alt=\"a\\cdot b\" align=\"absmiddle\" \/>. W ten oto spos\u00f3b otrzymamy kolejne wyra\u017cenie algebraiczne.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Przy zapisie wyra\u017cenia algebraicznego w postaci iloczynu nieraz mo\u017cemy pomin\u0105\u0107 znak mno\u017cenia. Zamiast <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1\\cdot&amp;space;x\\cdot&amp;space;y\" alt=\"1\\cdot x\\cdot y\" align=\"absmiddle\" \/>\u00a0zapiszemy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?xy\" alt=\"xy\" align=\"absmiddle\" \/> pomijaj\u0105c znaki mno\u017cenia oraz jedynk\u0119 lub\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3\\cdot&amp;space;a}{5}=\\frac{3a}{5}\" alt=\"\\frac{3\\cdot a}{5}=\\frac{3a}{5}\" align=\"absmiddle\" \/>. Jednak musimy pami\u0119ta\u0107, \u017ce chocia\u017c znaku mno\u017cenia nie wida\u0107, to jest to iloczyn!<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #4472c4;\">Wyra\u017cenie algebraiczne<\/span> jest <span style=\"color: #4472c4;\">uporz\u0105dkowane<\/span> kiedy w iloczynie wyst\u0119puje najpierw liczba, a p\u00f3\u017aniej litery, w kolejno\u015bci alfabetycznej:<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?9xz^{3},\\:&amp;space;0,25xz^{3},\\:&amp;space;\\frac{1}{7}xz^{3}\" alt=\"9xz^{3},\\: 0,25xz^{3},\\: \\frac{1}{7}xz^{3}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Je\u017celi, litery wyst\u0119puj\u0105ce w wyra\u017ceniu zast\u0105pimy liczbami, to po wykonaniu odpowiednich oblicze\u0144 otrzymamy <strong><span style=\"color: #4472c4;\">warto\u015b\u0107 wyra\u017cenia algebraicznego<\/span><\/strong>.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Obliczmy warto\u015b\u0107 wyra\u017cenia algebraicznego\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a+3b\" alt=\"a+3b\" align=\"absmiddle\" \/> dla <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a=1,\\:&amp;space;b=\\left&amp;space;(&amp;space;-1&amp;space;\\right&amp;space;)\" alt=\"a=1,\\: b=\\left ( -1 \\right )\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1+3\\cdot&amp;space;\\left&amp;space;(&amp;space;-1&amp;space;\\right&amp;space;)=1+\\left&amp;space;(&amp;space;-3&amp;space;\\right&amp;space;)=1-3=-2\" alt=\"1+3\\cdot \\left ( -1 \\right )=1+\\left ( -3 \\right )=1-3=-2\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Wyra\u017cenia algebraiczne nazywamy wyrazami <span style=\"color: #4472c4;\">podobnymi<\/span>, gdy w iloczynach wyst\u0119puj\u0105 litery, w tych samych pot\u0119gach, r\u00f3\u017cni\u0105ce si\u0119 jedynie liczbami np:<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?5a^{2}bc,\\:&amp;space;\\frac{3}{4}a^{2}bc,\\:&amp;space;0,33a^{2}bc\" alt=\"5a^{2}bc,\\: \\frac{3}{4}a^{2}bc,\\: 0,33a^{2}bc\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?9xz^{3},\\:&amp;space;0,25xz^{3},\\:&amp;space;\\frac{1}{7}xz^{3}\" alt=\"9xz^{3},\\: 0,25xz^{3},\\: \\frac{1}{7}xz^{3}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Je\u015bli ju\u017c pogrupujemy wyrazy podobne to mo\u017cemy wykonywa\u0107 dzia\u0142ania, na kilku wyra\u017ceniach algebraicznych. Pami\u0119tajcie o wykonywaniu dzia\u0142a\u0144 zgodnie z kolejno\u015bci\u0105. W taki spos\u00f3b, otrzymamy wyra\u017cenie algebraiczne, zapisane w prostszej postaci.<\/span><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a+b+c+4a-3b+2c=(1+4)a+(1-3)b+(1+2)c=\" alt=\"a+b+c+4a-3b+2c=(1+4)a+(1-3)b+(1+2)c=\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?=5a+(-2)b+3c=5a-2b+3c\" alt=\"=5a+(-2)b+3c=5a-2b+3c\" align=\"absmiddle\" \/><\/p>\n<p><span style=\"font-size: 14pt;\"><strong><span style=\"font-family: 'book antiqua', palatino, serif; color: #4472c4;\">R\u00f3wnania<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">R\u00f3wnanie to nic innego jak <\/span><span style=\"font-family: 'book antiqua', palatino, serif;\">przyr\u00f3wnane<\/span><span style=\"font-family: 'book antiqua', palatino, serif;\"> j<\/span><span style=\"font-family: 'book antiqua', palatino, serif;\">ednego wyra\u017cenie algebraicznego do innego wyra\u017cenia algebraicznego.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Aby <span style=\"color: #4472c4;\">rozwi\u0105za\u0107 dane r\u00f3wnanie<\/span> musimy znale\u017a\u0107 liczb\u0119, kt\u00f3ra je spe\u0142nia.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Znajd\u017a liczby spe\u0142niaj\u0105ce przyk\u0142adowe r\u00f3wnania:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a+6=10\" alt=\"a+6=10\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Aby znale\u017a\u0107 w\u0142a\u015bciwe rozwi\u0105zanie musimy odpowiedzie\u0107 na pytanie: Jak\u0105 liczb\u0119 nale\u017cy doda\u0107 do 6, aby otrzyma\u0107 10?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Odpowied\u017a nasuwa si\u0119 automatycznie: musi to by\u0107 liczba <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?4\" alt=\"4\" align=\"absmiddle\" \/>.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?b-5=2\" alt=\"b-5=2\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Patrz\u0105c na przyk\u0142ad powy\u017cej odpowiedzmy na pytanie: Od jakiej liczby musimy odj\u0105\u0107 5, \u017ceby otrzyma\u0107 2?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Oczywi\u015bcie od 7.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?10-c=2\" alt=\"10-c=2\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Analogicznie jak powy\u017cej jak\u0105 liczb\u0119 nale\u017cy odj\u0105\u0107 od 10, \u017ceby otrzyma\u0107 2?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Tak dok\u0142adnie, chodzi o 8.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?5\\cdot&amp;space;d=30\" alt=\"5\\cdot d=30\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Jaka liczb\u0119 nale\u017cy pomno\u017cy\u0107 przez 5, aby otrzyma\u0107 30?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Oczywi\u015bcie, jest tylko jedna mo\u017cliwo\u015b\u0107, mianowicie 6.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?e:10=4\" alt=\"e:10=4\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Jak\u0105 liczb\u0119 nale\u017cy podzieli\u0107 przez 10, aby otrzyma\u0107 wynik 4?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Oczywi\u015bcie, szukana liczba to 40.<\/span><\/p>\n<ul>\n<li style=\"text-align: left;\"><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?100:f=4\" alt=\"100:f=4\" align=\"absmiddle\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Przez jaka liczb\u0119 nale\u017cy podzieli\u0107 100, aby otrzyma\u0107 4?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Odpowied\u017a jest prosta. Szukana liczba to 25.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Prawda, \u017ce proste jest rozwi\u0105zywanie nieskomplikowanych r\u00f3wna\u0144. W wielu przypadkach mo\u017cna od razu poda\u0107 wynik. Niestety, nie zawsze jest tak \u0142atwo.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Pami\u0119tajmy, \u017ce r\u00f3wnanie nie zmieni swojego rozwi\u0105zania je\u015bli dodamy, odejmiemy, pomno\u017cymy lub podzielimy obie jego strony przez t\u0119 sam\u0105 liczb\u0119. Poni\u017cej przedstawiam kilka sposob\u00f3w dzi\u0119ki, kt\u00f3rym nawet skomplikowane r\u00f3wnanie stanie si\u0119 \u0142atwe do rozwi\u0105zania.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1164 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodaw-300x124.png\" alt=\"\" width=\"319\" height=\"132\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodaw-300x124.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodaw.png 432w\" sizes=\"(max-width: 319px) 100vw, 319px\" \/><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\" wp-image-1169 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-odejm-300x132.png\" alt=\"\" width=\"316\" height=\"139\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-odejm-300x132.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-odejm.png 411w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1165 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawanie-300x108.png\" alt=\"\" width=\"344\" height=\"124\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawanie-300x108.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawanie.png 452w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Strony r\u00f3wnania mo\u017cemy zamienia\u0107 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=6\" alt=\"x=6\" align=\"absmiddle\" \/> to jest to samo co <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?6=x\" alt=\"6=x\" align=\"absmiddle\" \/>.<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1166 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawaniee-300x117.png\" alt=\"\" width=\"330\" height=\"128\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawaniee-300x117.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dodawaniee.png 438w\" sizes=\"(max-width: 330px) 100vw, 330px\" \/><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1167 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-dziel.png\" alt=\"\" width=\"221\" height=\"211\" \/><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1168 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-mno\u017c-300x216.png\" alt=\"\" width=\"246\" height=\"177\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-mno\u017c-300x216.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/r\u00f3wnania-mno\u017c.png 317w\" sizes=\"(max-width: 246px) 100vw, 246px\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Jest jeszcze jedna rzecz, kt\u00f3ra jest cz\u0119sto pomijana przez uczni\u00f3w i jest to b\u0142\u0105d. Mianowicie, je\u015bli ju\u017c znale\u017ali\u015bmy rozwi\u0105zanie r\u00f3wnania, to nale\u017cy <span style=\"text-decoration: underline;\">sprawdzi\u0107 czy jest to rozwi\u0105zanie prawid\u0142owe<\/span>. Aby to zrobi\u0107, wstawiamy nasz wynik do jednej strony r\u00f3wnania pocz\u0105tkowego i sprawdzamy czy wynik pokrywa si\u0119 z drug\u0105 stron\u0105 tego r\u00f3wnania.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Znajd\u017amy x:<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}x=12\" alt=\"\\frac{3}{4}x=12\" align=\"absmiddle\" \/>\u00a0 (mno\u017cymy obie strony r\u00f3wnania przez\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{3}\" alt=\"\\frac{4}{3}\" align=\"absmiddle\" \/>, b\u0105d\u017a dzielimy przez <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{4}\" alt=\"\\frac{3}{4}\" align=\"absmiddle\" \/> to ju\u017c jak komu\u015b wygodnie)<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{3}\\cdot&amp;space;\\frac{3}{4}x=\\frac{4}{3}\\cdot&amp;space;12\" alt=\"\\frac{4}{3}\\cdot \\frac{3}{4}x=\\frac{4}{3}\\cdot 12\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=16\" alt=\"x=16\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">A teraz sprawdzamy czy dobrze wyliczyli\u015bmy x:<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?L=\\frac{3}{4}\\cdot&amp;space;16=12=P\" alt=\"L=\\frac{3}{4}\\cdot 16=12=P\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Czyli dobrze wykonali\u015bmy obliczenia. Szukana liczba to <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=16\" alt=\"x=16\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"text-decoration: underline;\"><span style=\"font-family: 'book antiqua', palatino, serif; color: #4472c4; text-decoration: underline;\">Aby poprawnie rozwi\u0105za\u0107 zadanie za pomoc\u0105 r\u00f3wna\u0144 nale\u017cy si\u0119 stosowa\u0107 do kilku zasad:<\/span><\/span><\/p>\n<ol style=\"list-style-type: upper-roman;\">\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">Ustalamy co jest niewiadom\u0105.<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">Zapisujemy r\u00f3wnanie, kt\u00f3re wynika z tre\u015bci.<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">Rozwi\u0105zujemy r\u00f3wnanie.<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">Sprawdzamy czy nasze rozwi\u0105zanie spe\u0142nia r\u00f3wnanie.<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">zapisujemy odpowied\u017a.<\/span><\/li>\n<\/ol>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Je\u015bli b\u0119dziemy si\u0119 stosowa\u0107 do tych regu\u0142, to \u017cadne zadanie nie b\u0119dzie trudne do rozwi\u0105zania.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wyra\u017cenia algebraiczne Witamy w algebrze!!! Od tego momentu b\u0119dzie to nieodzowna cz\u0119\u015b\u0107 matematyki, w kt\u00f3rej du\u017co &#8222;pracujemy&#8221; na literkach. Niejednokrotnie, w dotychczasowej nauce matematyki, korzystali\u015bcie z wyra\u017ce\u0144 algebraicznych. By\u0142y one dost\u0119pne m.in. na geometrii, w postaci wzor\u00f3w do obliczania pola czy obwodu. Najprostszymi wyra\u017ceniami algebraicznymi s\u0105: litery (), liczby () oraz po\u0142\u0105czone dzia\u0142aniami litery z [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,13],"tags":[],"class_list":["post-1158","post","type-post","status-publish","format-standard","hentry","category-baza-wiedzy","category-klasa-vi"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Wyra\u017cenia algebraiczne. 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