Whakaoti i te {l}{4x+y=-7}{2x+6y=-11}

$$\left. \begin{array} { l } { 4 x + y = - 7 } \\ { 2 x + 6 y = - 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{31}{22} = -1\frac{9}{22} \approx -1.409090909$$

$$y = -\frac{15}{11} = -1\frac{4}{11} \approx -1.363636364$$

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.

$$4x+y=-7,2x+6y=-11$$

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

$$4x+y=-7$$

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

$$4x=-y-7$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

Whakakapia te \frac{-y-7}{4} mō te x ki tērā atu whārite, 2x+6y=-11.

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

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

$$-\frac{1}{2}y-\frac{7}{2}+6y=-11$$

Tāpiri -\frac{y}{2} ki te 6y.

$$\frac{11}{2}y-\frac{7}{2}=-11$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$\frac{11}{2}y=-\frac{15}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{15}{11}$$

Whakaurua te -\frac{15}{11} mō y ki x=-\frac{1}{4}y-\frac{7}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\left(-\frac{15}{11}\right)-\frac{7}{4}$$

Whakareatia -\frac{1}{4} ki te -\frac{15}{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.

$$x=\frac{15}{44}-\frac{7}{4}$$

Tāpiri -\frac{7}{4} ki te \frac{15}{44} 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.

$$x=-\frac{31}{22}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{31}{22},y=-\frac{15}{11}$$

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.

$$4x+y=-7,2x+6y=-11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\2&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\2&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\2&6\end{matrix}\right))\left(\begin{matrix}4&1\\2&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&6\end{matrix}\right))\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&6\end{matrix}\right))\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&6\end{matrix}\right))\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{4\times 6-2}&-\frac{1}{4\times 6-2}\\-\frac{2}{4\times 6-2}&\frac{4}{4\times 6-2}\end{matrix}\right)\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&-\frac{1}{22}\\-\frac{1}{11}&\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\left(-7\right)-\frac{1}{22}\left(-11\right)\\-\frac{1}{11}\left(-7\right)+\frac{2}{11}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{31}{22}\\-\frac{15}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{31}{22},y=-\frac{15}{11}$$

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.

$$4x+y=-7,2x+6y=-11$$

Kia ōrite ai a 4x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$2\times 4x+2y=2\left(-7\right),4\times 2x+4\times 6y=4\left(-11\right)$$

Whakarūnātia.

$$8x+2y=-14,8x+24y=-44$$

Me tango 8x+24y=-44 mai i 8x+2y=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+2y-24y=-14+44$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-24y=-14+44$$

Tāpiri 2y ki te -24y.

$$-22y=-14+44$$

Tāpiri -14 ki te 44.

$$-22y=30$$

Whakawehea ngā taha e rua ki te -22.

$$y=-\frac{15}{11}$$

Whakaurua te -\frac{15}{11} mō y ki 2x+6y=-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+6\left(-\frac{15}{11}\right)=-11$$

Whakareatia 6 ki te -\frac{15}{11}.

$$2x-\frac{90}{11}=-11$$

Me tāpiri \frac{90}{11} ki ngā taha e rua o te whārite.

$$2x=-\frac{31}{11}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{31}{22}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{31}{22},y=-\frac{15}{11}$$