>Likninger>Likningssystem (addisjonsmetoden)

Likningssystem (addisjonsmetoden)

Nå skal du lære siste metode for å løse likningssystem. Nedenfor ser du en oppskrift for hvordan man går frem. Under ser du et eksempel på hvordan man løser likningssett med denne metoden.

Regel

Addisjonsmetoden

1.: Velg en av variablene som du vil bli kvitt.
2.: Multipliser likningene $I$ og $I I$ med de tallene som gjør at variabelen du har valgt får samme konstant, men motsatt fortegn.
3.: Skriv opp de nye likningene under de to andre.
4.: Adder likning $I I$ med $I$ og skriv opp svaret under disse likningene. Du har nå én likning med én ukjent. Løs den.
5.: Sett svaret du finner, inn i likning $I$ og løs ut den siste variabelen.
6.: Skriv svaret på koordinatform: $SVAR: (x, y) = (a, b)$

Eksempel 1

Løs likningssettet

\begin{array}{l} I1. Vi         bestemmer         oss         for         å         bli         kvitt y . 2. Siden du har 2 y i andre         linje og y i øverste         linje, må du gange y med - 2 for å bli         kvitt y -verdien.         I dette tilfellet trenger du ikke gange likning I I med noen faktor: \begin{array}{l} I y + 2 x = 1 & | \cdot (- 2) \\ I I \underset{̲}{2 y - x = 2} \end{array} 3. Skriv opp de to likningene en gang til etter at endringene er gjort. \begin{array}{l} - 2 y - 4 x & = - 2 \\ \underset{̲}{2 y - x} & \underset{̲}{= 2} \end{array} 4. Adder de to likningene så du blir kvitt y, og løs         for x : \begin{array}{l} - 2 y + 2 y - 4 x - x & = - 2 + 2 \\ 0 y - 5 x & = 0 \\ x & = 0 \end{array} 5. Sett svaret inn i en av likningene I eller I I .         Her kan du velge hvilken du vil. Jeg velger likning I : \begin{array}{l} y + 2 x = 1 \\ y + 2 \cdot (0) = 1 \\ y + 0 = 1 \\ y = 1 \end{array} 6. Skriv svaret på koordinatform: SVAR: (x, y) = (0, 1)Vil du vite mer? Registrer deg Det er gratis!Forrige oppslag Likningssystem (innsetting) Neste oppslag Aktiviteter 9Hjem All matte Spill Progresjon LigaOm oss Om House of Math Om ansatte Karriere Media Foredrag Blogg Kontakt support@houseofmath.com Booking av privatundervisning +47 22 150 300 Adresse Bygdøy allé 23
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Under ser du et eksempel på hvordan man løser likningssett med denne metoden. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-179\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Regel\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eAddisjonsmetoden\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Velg         en         av         variablene         som         du         vil         bli         kvitt.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Multipliser         likningene         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e       og         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e       med         de         tallene         som         gjør         at         variabelen         du         har         valgt         får         samme         konstant,         men         motsatt         fortegn.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Skriv         opp         de         nye         likningene         under         de         to         andre.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     4.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Adder         likning         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e       med         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e       og         skriv         opp         svaret         under         disse         likningene.         Du         har         nå         én         likning         med         én         ukjent.         Løs         den.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     5.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Sett         svaret         du         finner,         inn         i         likning         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e       og         løs         ut         den         siste         variabelen.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     6.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Skriv         svaret         på         koordinatform:         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003eSVAR: \u003c/mtext\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi  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stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                                   \u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Vi         bestemmer         oss         for         å         bli         kvitt         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math type=\"punct\"\u003e.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSiden du har \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e i andre         linje og \u003cmath 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\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmo  accent=\"true\"\u003e̲\u003c/mo\u003e\u003c/munder\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003cmunder accentunder=\"false\"  class=\"mml-underline\"\u003e\u003cmrow\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmphantom\u003e  \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003c/mphantom\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo  accent=\"true\"\u003e̲\u003c/mo\u003e\u003c/munder\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     4.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eAdder de to likningene så du blir kvitt         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e, og løs         for \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math type=\"divide\"\u003e:         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                                \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmphantom\u003e  \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003c/mphantom\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  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xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e         eller \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eI\u003c/mi\u003e\u003cmi  \u003eI\u003c/mi\u003e\u003c/math\u003e.         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width=\"2em\"/\u003e\u003c/mtd\u003e                                         \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                         \u003cmtd  class=\"align-label\"\u003e                                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/mtd\u003e                                                \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                         \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                         \u003cmtd  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