Whakaoti i te {l}{x-2y=-11}{3x+7y=32}

$$\left. \begin{array} { l } { x - 2 y = - 11 } \\ { 3 x + 7 y = 32 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=5$$

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.

$$x-2y=-11,3x+7y=32$$

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

$$x-2y=-11$$

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

$$x=2y-11$$

Whakakapia te 2y-11 mō te x ki tērā atu whārite, 3x+7y=32.

$$3\left(2y-11\right)+7y=32$$

Whakareatia 3 ki te 2y-11.

$$6y-33+7y=32$$

Tāpiri 6y ki te 7y.

$$13y-33=32$$

Me tāpiri 33 ki ngā taha e rua o te whārite.

$$13y=65$$

Whakawehea ngā taha e rua ki te 13.

$$y=5$$

Whakaurua te 5 mō y ki x=2y-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times 5-11$$

Whakareatia 2 ki te 5.

$$x=10-11$$

Tāpiri -11 ki te 10.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$

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.

$$x-2y=-11,3x+7y=32$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\32\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\3&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\3&7\end{matrix}\right))\left(\begin{matrix}1&-2\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&7\end{matrix}\right))\left(\begin{matrix}-11\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&7\end{matrix}\right))\left(\begin{matrix}-11\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&7\end{matrix}\right))\left(\begin{matrix}-11\\32\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-2\times 3\right)}&-\frac{-2}{7-\left(-2\times 3\right)}\\-\frac{3}{7-\left(-2\times 3\right)}&\frac{1}{7-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-11\\32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}&\frac{2}{13}\\-\frac{3}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}-11\\32\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}\left(-11\right)+\frac{2}{13}\times 32\\-\frac{3}{13}\left(-11\right)+\frac{1}{13}\times 32\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=5$$

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.

$$x-2y=-11,3x+7y=32$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\left(-2\right)y=3\left(-11\right),3x+7y=32$$

Whakarūnātia.

$$3x-6y=-33,3x+7y=32$$

Me tango 3x+7y=32 mai i 3x-6y=-33 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-6y-7y=-33-32$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-7y=-33-32$$

Tāpiri -6y ki te -7y.

$$-13y=-33-32$$

Tāpiri -33 ki te -32.

$$-13y=-65$$

Whakawehea ngā taha e rua ki te -13.

$$y=5$$

Whakaurua te 5 mō y ki 3x+7y=32. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+7\times 5=32$$

Whakareatia 7 ki te 5.

$$3x+35=32$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$3x=-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$