Whakaoti i te {l}{2x+8y=64}{7x+y=8}

$$\left. \begin{array} { l } { 2 x + 8 y = 64 } \\ { 7 x + y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=8$$

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.

$$2x+8y=64,7x+y=8$$

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

$$2x+8y=64$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$2x=-8y+64$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-8y+64\right)$$

Whakareatia \frac{1}{2} ki te -8y+64.

$$x=-4y+32$$

Whakakapia te -4y+32 mō te x ki tērā atu whārite, 7x+y=8.

$$7\left(-4y+32\right)+y=8$$

Whakareatia 7 ki te -4y+32.

$$-28y+224+y=8$$

Tāpiri -28y ki te y.

$$-27y+224=8$$

Me tango 224 mai i ngā taha e rua o te whārite.

$$-27y=-216$$

Whakawehea ngā taha e rua ki te -27.

$$y=8$$

Whakaurua te 8 mō y ki x=-4y+32. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times 8+32$$

Whakareatia -4 ki te 8.

$$x=-32+32$$

Tāpiri 32 ki te -32.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=8$$

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.

$$2x+8y=64,7x+y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&8\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}64\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&8\\7&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&8\\7&1\end{matrix}\right))\left(\begin{matrix}2&8\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\7&1\end{matrix}\right))\left(\begin{matrix}64\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\7&1\end{matrix}\right))\left(\begin{matrix}64\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&8\\7&1\end{matrix}\right))\left(\begin{matrix}64\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-8\times 7}&-\frac{8}{2-8\times 7}\\-\frac{7}{2-8\times 7}&\frac{2}{2-8\times 7}\end{matrix}\right)\left(\begin{matrix}64\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{54}&\frac{4}{27}\\\frac{7}{54}&-\frac{1}{27}\end{matrix}\right)\left(\begin{matrix}64\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{54}\times 64+\frac{4}{27}\times 8\\\frac{7}{54}\times 64-\frac{1}{27}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=8$$

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.

$$2x+8y=64,7x+y=8$$

Kia ōrite ai a 2x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$7\times 2x+7\times 8y=7\times 64,2\times 7x+2y=2\times 8$$

Whakarūnātia.

$$14x+56y=448,14x+2y=16$$

Me tango 14x+2y=16 mai i 14x+56y=448 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+56y-2y=448-16$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$56y-2y=448-16$$

Tāpiri 56y ki te -2y.

$$54y=448-16$$

Tāpiri 448 ki te -16.

$$54y=432$$

Whakawehea ngā taha e rua ki te 54.

$$y=8$$

Whakaurua te 8 mō y ki 7x+y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+8=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$7x=0$$

Whakawehea ngā taha e rua ki te 7.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=8$$