Whakaoti i te {l}{-x-6y=-16}{5x-y=18}

$$\left. \begin{array} { l } { - x - 6 y = - 16 } \\ { 5 x - y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=2$$

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.

$$-x-6y=-16,5x-y=18$$

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

$$-x-6y=-16$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$-x=6y-16$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(6y-16\right)$$

Whakareatia -1 ki te 6y-16.

$$x=-6y+16$$

Whakakapia te -6y+16 mō te x ki tērā atu whārite, 5x-y=18.

$$5\left(-6y+16\right)-y=18$$

Whakareatia 5 ki te -6y+16.

$$-30y+80-y=18$$

Tāpiri -30y ki te -y.

$$-31y+80=18$$

Me tango 80 mai i ngā taha e rua o te whārite.

$$-31y=-62$$

Whakawehea ngā taha e rua ki te -31.

$$y=2$$

Whakaurua te 2 mō y ki x=-6y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6\times 2+16$$

Whakareatia -6 ki te 2.

$$x=-12+16$$

Tāpiri 16 ki te -12.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

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.

$$-x-6y=-16,5x-y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right))\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right))\left(\begin{matrix}-16\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right))\left(\begin{matrix}-16\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-6\\5&-1\end{matrix}\right))\left(\begin{matrix}-16\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-\left(-1\right)-\left(-6\times 5\right)}&-\frac{-6}{-\left(-1\right)-\left(-6\times 5\right)}\\-\frac{5}{-\left(-1\right)-\left(-6\times 5\right)}&-\frac{1}{-\left(-1\right)-\left(-6\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-16\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{31}&\frac{6}{31}\\-\frac{5}{31}&-\frac{1}{31}\end{matrix}\right)\left(\begin{matrix}-16\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{31}\left(-16\right)+\frac{6}{31}\times 18\\-\frac{5}{31}\left(-16\right)-\frac{1}{31}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

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.

$$-x-6y=-16,5x-y=18$$

Kia ōrite ai a -x 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 -1.

$$5\left(-1\right)x+5\left(-6\right)y=5\left(-16\right),-5x-\left(-y\right)=-18$$

Whakarūnātia.

$$-5x-30y=-80,-5x+y=-18$$

Me tango -5x+y=-18 mai i -5x-30y=-80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x-30y-y=-80+18$$

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-30y-y=-80+18$$

Tāpiri -30y ki te -y.

$$-31y=-80+18$$

Tāpiri -80 ki te 18.

$$-31y=-62$$

Whakawehea ngā taha e rua ki te -31.

$$y=2$$

Whakaurua te 2 mō y ki 5x-y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2=18$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$