Whakaoti i te {l}{4x+7y=2}{5x+6y=4}

$$\left. \begin{array} { l } { 4 x + 7 y = 2 } \\ { 5 x + 6 y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{16}{11} = 1\frac{5}{11} \approx 1.454545455$$

$$y=-\frac{6}{11}\approx -0.545454545$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$4x+7y=2,5x+6y=4$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$4x+7y=2$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$4x=-7y+2$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-7y+2\right)$$

Whakareatia \frac{1}{4} ki te -7y+2.

$$x=-\frac{7}{4}y+\frac{1}{2}$$

Whakakapia te -\frac{7y}{4}+\frac{1}{2} mō te x ki tērā atu whārite, 5x+6y=4.

$$5\left(-\frac{7}{4}y+\frac{1}{2}\right)+6y=4$$

Whakareatia 5 ki te -\frac{7y}{4}+\frac{1}{2}.

$$-\frac{35}{4}y+\frac{5}{2}+6y=4$$

Tāpiri -\frac{35y}{4} ki te 6y.

$$-\frac{11}{4}y+\frac{5}{2}=4$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$-\frac{11}{4}y=\frac{3}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{6}{11}$$

Whakaurua te -\frac{6}{11} mō y ki x=-\frac{7}{4}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{4}\left(-\frac{6}{11}\right)+\frac{1}{2}$$

Whakareatia -\frac{7}{4} ki te -\frac{6}{11} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{21}{22}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{21}{22} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{16}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{11},y=-\frac{6}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$4x+7y=2,5x+6y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&7\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&7\\5&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&7\\5&6\end{matrix}\right))\left(\begin{matrix}4&7\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&6\end{matrix}\right))\left(\begin{matrix}2\\4\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&6\end{matrix}\right))\left(\begin{matrix}2\\4\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&6\end{matrix}\right))\left(\begin{matrix}2\\4\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{4\times 6-7\times 5}&-\frac{7}{4\times 6-7\times 5}\\-\frac{5}{4\times 6-7\times 5}&\frac{4}{4\times 6-7\times 5}\end{matrix}\right)\left(\begin{matrix}2\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{11}&\frac{7}{11}\\\frac{5}{11}&-\frac{4}{11}\end{matrix}\right)\left(\begin{matrix}2\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{11}\times 2+\frac{7}{11}\times 4\\\frac{5}{11}\times 2-\frac{4}{11}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{11}\\-\frac{6}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{11},y=-\frac{6}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$4x+7y=2,5x+6y=4$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\times 7y=5\times 2,4\times 5x+4\times 6y=4\times 4$$

Whakarūnātia.

$$20x+35y=10,20x+24y=16$$

Me tango 20x+24y=16 mai i 20x+35y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+35y-24y=10-16$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35y-24y=10-16$$

Tāpiri 35y ki te -24y.

$$11y=10-16$$

Tāpiri 10 ki te -16.

$$11y=-6$$

Whakawehea ngā taha e rua ki te 11.

$$y=-\frac{6}{11}$$

Whakaurua te -\frac{6}{11} mō y ki 5x+6y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+6\left(-\frac{6}{11}\right)=4$$

Whakareatia 6 ki te -\frac{6}{11}.

$$5x-\frac{36}{11}=4$$

Me tāpiri \frac{36}{11} ki ngā taha e rua o te whārite.

$$5x=\frac{80}{11}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{16}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{11},y=-\frac{6}{11}$$