Whakaoti i te {l}{-5x+y=-11}{4x-6y=14}

$$\left. \begin{array}{l}{ - 5 x + y = - 11 }\\{ 4 x - 6 y = 14 }\end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-1$$

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.

$$-5x+y=-11,4x-6y=14$$

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

$$-5x+y=-11$$

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

$$-5x=-y-11$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-y-11\right)$$

Whakareatia -\frac{1}{5} ki te -y-11.

$$x=\frac{1}{5}y+\frac{11}{5}$$

Whakakapia te \frac{11+y}{5} mō te x ki tērā atu whārite, 4x-6y=14.

$$4\left(\frac{1}{5}y+\frac{11}{5}\right)-6y=14$$

Whakareatia 4 ki te \frac{11+y}{5}.

$$\frac{4}{5}y+\frac{44}{5}-6y=14$$

Tāpiri \frac{4y}{5} ki te -6y.

$$-\frac{26}{5}y+\frac{44}{5}=14$$

Me tango \frac{44}{5} mai i ngā taha e rua o te whārite.

$$-\frac{26}{5}y=\frac{26}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{26}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{1}{5}y+\frac{11}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\left(-1\right)+\frac{11}{5}$$

Whakareatia \frac{1}{5} ki te -1.

$$x=\frac{-1+11}{5}$$

Tāpiri \frac{11}{5} ki te -\frac{1}{5} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$

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.

$$-5x+y=-11,4x-6y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&1\\4&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right))\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right))\left(\begin{matrix}-11\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right))\left(\begin{matrix}-11\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\4&-6\end{matrix}\right))\left(\begin{matrix}-11\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-5\left(-6\right)-4}&-\frac{1}{-5\left(-6\right)-4}\\-\frac{4}{-5\left(-6\right)-4}&-\frac{5}{-5\left(-6\right)-4}\end{matrix}\right)\left(\begin{matrix}-11\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{13}&-\frac{1}{26}\\-\frac{2}{13}&-\frac{5}{26}\end{matrix}\right)\left(\begin{matrix}-11\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{13}\left(-11\right)-\frac{1}{26}\times 14\\-\frac{2}{13}\left(-11\right)-\frac{5}{26}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-1$$

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.

$$-5x+y=-11,4x-6y=14$$

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

$$4\left(-5\right)x+4y=4\left(-11\right),-5\times 4x-5\left(-6\right)y=-5\times 14$$

Whakarūnātia.

$$-20x+4y=-44,-20x+30y=-70$$

Me tango -20x+30y=-70 mai i -20x+4y=-44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-20x+20x+4y-30y=-44+70$$

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.

$$4y-30y=-44+70$$

Tāpiri 4y ki te -30y.

$$-26y=-44+70$$

Tāpiri -44 ki te 70.

$$-26y=26$$

Whakawehea ngā taha e rua ki te -26.

$$y=-1$$

Whakaurua te -1 mō y ki 4x-6y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-6\left(-1\right)=14$$

Whakareatia -6 ki te -1.

$$4x+6=14$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-1$$