Whakaoti i te {l}{-x-3y=12}{-5x-9y=18}

$$\left. \begin{array} { l } { - x - 3 y = 12 } \\ { - 5 x - 9 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=-7$$

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.

$$-x-3y=12,-5x-9y=18$$

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

$$-x-3y=12$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$-x=3y+12$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(3y+12\right)$$

Whakareatia -1 ki te 12+3y.

$$x=-3y-12$$

Whakakapia te -3y-12 mō te x ki tērā atu whārite, -5x-9y=18.

$$-5\left(-3y-12\right)-9y=18$$

Whakareatia -5 ki te -3y-12.

$$15y+60-9y=18$$

Tāpiri 15y ki te -9y.

$$6y+60=18$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$6y=-42$$

Whakawehea ngā taha e rua ki te 6.

$$y=-7$$

Whakaurua te -7 mō y ki x=-3y-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\left(-7\right)-12$$

Whakareatia -3 ki te -7.

$$x=21-12$$

Tāpiri -12 ki te 21.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-7$$

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.

$$-x-3y=12,-5x-9y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right))\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right))\left(\begin{matrix}12\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right))\left(\begin{matrix}12\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-3\\-5&-9\end{matrix}\right))\left(\begin{matrix}12\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-\left(-9\right)-\left(-3\left(-5\right)\right)}&-\frac{-3}{-\left(-9\right)-\left(-3\left(-5\right)\right)}\\-\frac{-5}{-\left(-9\right)-\left(-3\left(-5\right)\right)}&-\frac{1}{-\left(-9\right)-\left(-3\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}12\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}&-\frac{1}{2}\\-\frac{5}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}12\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\times 12-\frac{1}{2}\times 18\\-\frac{5}{6}\times 12+\frac{1}{6}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=-7$$

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.

$$-x-3y=12,-5x-9y=18$$

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.

$$-5\left(-1\right)x-5\left(-3\right)y=-5\times 12,-\left(-5\right)x-\left(-9y\right)=-18$$

Whakarūnātia.

$$5x+15y=-60,5x+9y=-18$$

Me tango 5x+9y=-18 mai i 5x+15y=-60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+15y-9y=-60+18$$

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.

$$15y-9y=-60+18$$

Tāpiri 15y ki te -9y.

$$6y=-60+18$$

Tāpiri -60 ki te 18.

$$6y=-42$$

Whakawehea ngā taha e rua ki te 6.

$$y=-7$$

Whakaurua te -7 mō y ki -5x-9y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-9\left(-7\right)=18$$

Whakareatia -9 ki te -7.

$$-5x+63=18$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$-5x=-45$$

Whakawehea ngā taha e rua ki te -5.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-7$$