{"id":2333,"date":"2025-05-20T20:39:33","date_gmt":"2025-05-20T17:39:33","guid":{"rendered":"https:\/\/cedra.academy\/?p=2333"},"modified":"2025-09-09T21:00:29","modified_gmt":"2025-09-09T18:00:29","slug":"rezolvarea-ecuatiilor-si-inecuatiilor-liniare-cu-o-necunoscuta-si-aplicarea-lor-in-probleme-practice","status":"publish","type":"post","link":"https:\/\/cedra.academy\/?p=2333","title":{"rendered":"Rezolvarea ecua\u021biilor \u0219i inecua\u021biilor liniare cu o necunoscut\u0103 \u0219i aplicarea lor \u00een probleme practice"},"content":{"rendered":"\n
Ecua\u021biile \u0219i inecua\u021biile liniare PPTX .<\/a>Descarc\u0103<\/a><\/div>\n\n\n\n

<\/p>\n\n\n\n

1. Ecua\u021biile liniare cu o necunoscut\u0103<\/h3>\n\n\n\n

O ecua\u021bie liniar\u0103 cu o necunoscut\u0103<\/strong> este o egalitate care con\u021bine o singur\u0103 variabil\u0103, notat\u0103 de obicei cu xxx, \u0219i poate fi scris\u0103 \u00een forma general\u0103: ax+b=0ax + b = 0ax+b=0<\/p>\n\n\n\n

unde aaa \u0219i bbb sunt numere reale, iar a\u22600a \\neq 0a\ue020=0.<\/p>\n\n\n\n

Exemplu:<\/strong> 3x+5=03x + 5 = 03x+5=0<\/p>\n\n\n\n

1.1. Metoda de rezolvare<\/h4>\n\n\n\n

Pentru a rezolva o ecua\u021bie liniar\u0103, urm\u0103m pa\u0219ii:<\/p>\n\n\n\n

    \n
  1. Mut\u0103m termenii liberi pe cealalt\u0103 parte a egalit\u0103\u021bii:<\/li>\n<\/ol>\n\n\n\n

    3x=\u221253x = -53x=\u22125<\/p>\n\n\n\n

      \n
    1. \u00cemp\u0103r\u021bim ambele p\u0103r\u021bi la coeficientul necunoscutei:<\/li>\n<\/ol>\n\n\n\n

      x=\u221253x = -\\frac{5}{3}x=\u221235\u200b<\/p>\n\n\n\n

      Astfel, solu\u021bia ecua\u021biei este x=\u221253x = -\\frac{5}{3}x=\u221235\u200b.<\/p>\n\n\n\n

      2. Inecua\u021biile liniare cu o necunoscut\u0103<\/h3>\n\n\n\n

      O inecua\u021bie liniar\u0103 cu o necunoscut\u0103<\/strong> are forma: ax+b>0sauax+b<0sauax+b\u22650sauax+b\u22640ax + b > 0 \\quad \\text{sau} \\quad ax + b < 0 \\quad \\text{sau} \\quad ax + b \\ge 0 \\quad \\text{sau} \\quad ax + b \\le 0ax+b>0sauax+b<0sauax+b\u22650sauax+b\u22640<\/p>\n\n\n\n

      unde a\u22600a \\neq 0a\ue020=0.<\/p>\n\n\n\n

      Exemplu:<\/strong> 2x\u22124<02x – 4 < 02x\u22124<0<\/p>\n\n\n\n

      2.1. Metoda de rezolvare<\/h4>\n\n\n\n

      Pa\u0219ii sunt similari cu cei pentru ecua\u021bii:<\/p>\n\n\n\n

        \n
      1. Mut\u0103m termenul liber pe cealalt\u0103 parte:<\/li>\n<\/ol>\n\n\n\n

        2x<42x < 42x<4<\/p>\n\n\n\n

          \n
        1. \u00cemp\u0103r\u021bim ambele p\u0103r\u021bi la coeficientul lui xxx (pozitiv sau negativ \u2013 dac\u0103 e negativ, schimb\u0103m sensul inegalit\u0103\u021bii):<\/li>\n<\/ol>\n\n\n\n

          x<2x < 2x<2<\/p>\n\n\n\n

          Deci solu\u021bia este mul\u021bimea numerelor reale mai mici dec\u00e2t 2: x\u2208(\u2212\u221e,2)x \\in (-\\infty, 2)x\u2208(\u2212\u221e,2).<\/p>\n\n\n\n

          3. Aplicarea ecua\u021biilor \u0219i inecua\u021biilor \u00een probleme practice<\/h3>\n\n\n\n

          Ecua\u021biile \u0219i inecua\u021biile liniare sunt instrumente esen\u021biale pentru rezolvarea problemelor din via\u021ba real\u0103, cum ar fi:<\/p>\n\n\n\n

            \n
          1. Probleme cu bani:<\/strong>
            Exemplu: \u201eMaria are cu 5 lei mai mult dec\u00e2t Ana. Dac\u0103 Ana are xxx lei, c\u00e2\u021bi bani are Maria?\u201d
            Ecua\u021bia: x+5=yx + 5 = yx+5=y, unde yyy este suma Mariei.<\/li>\n\n\n\n
          2. Probleme de propor\u021bii \u0219i m\u0103sur\u0103tori:<\/strong>
            Exemplu: \u201eLungimea unui dreptunghi este cu 3 m mai mare dec\u00e2t l\u0103\u021bimea sa. Perimetrul dreptunghiului este 22 m. Care sunt dimensiunile dreptunghiului?\u201d
            Ecua\u021bia: 2(l+L)=222(l + L) = 222(l+L)=22
            Substituind L=l+3L = l + 3L=l+3, ob\u021binem:<\/li>\n<\/ol>\n\n\n\n

            2(l+l+3)=22\u21d24l+6=22\u21d2l=4 m,\u2009L=7 m2(l + l + 3) = 22 \\Rightarrow 4l + 6 = 22 \\Rightarrow l = 4 \\text{ m}, \\, L = 7 \\text{ m}2(l+l+3)=22\u21d24l+6=22\u21d2l=4 m,L=7 m<\/p>\n\n\n\n

              \n
            1. Probleme cu inecua\u021bii (condi\u021bii limit\u0103):<\/strong>
              Exemplu: \u201eO fabric\u0103 poate produce cel mult 500 de produse pe zi. Dac\u0103 xxx este num\u0103rul produselor produse, care este condi\u021bia pentru xxx?\u201d
              Inecuatia: x\u2264500x \\le 500x\u2264500<\/li>\n<\/ol>\n\n\n\n

              Astfel, elevii pot folosi ecua\u021biile \u0219i inecua\u021biile pentru a g\u0103si solu\u021bii \u0219i pentru a verifica dac\u0103 anumite condi\u021bii sunt respectate \u00een situa\u021bii reale.<\/p>\n\n\n\n

              4. Concluzii<\/h3>\n\n\n\n
                \n
              • Ecua\u021biile liniare cu o necunoscut\u0103 ne ajut\u0103 s\u0103 afl\u0103m valoarea unei variabile necunoscute.<\/li>\n\n\n\n
              • Inecua\u021biile liniare ne arat\u0103 condi\u021biile sub care anumite situa\u021bii sunt posibile.<\/li>\n\n\n\n
              • \u00cen via\u021ba real\u0103, aceste instrumente matematice se aplic\u0103 \u00een finan\u021be, m\u0103sur\u0103tori, planificarea produc\u021biei, programe de lucru \u0219i multe alte domenii.<\/li>\n<\/ul>\n\n\n\n
                \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                1. Ecua\u021biile liniare cu o necunoscut\u0103 O ecua\u021bie liniar\u0103 cu o necunoscut\u0103 este o egalitate care con\u021bine o singur\u0103 variabil\u0103, notat\u0103 de obicei cu xxx, \u0219i poate fi scris\u0103 \u00een […]<\/p>\n","protected":false},"author":1,"featured_media":2335,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,10],"tags":[],"_links":{"self":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2333"}],"collection":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2333"}],"version-history":[{"count":3,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2333\/revisions"}],"predecessor-version":[{"id":2339,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2333\/revisions\/2339"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/media\/2335"}],"wp:attachment":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2333"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2333"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}