Whakaoti i te {l}{5x-2y=12}{2x+y=13}

$$\left.\begin{array} { l } { 5 x - 2 y = 12 } \\ { 2 x + y = 13 } \end{array} \right\}$$

Whakaoti mō x, y

$$x = \frac{38}{9} = 4\frac{2}{9} \approx 4.222222222$$

$$y = \frac{41}{9} = 4\frac{5}{9} \approx 4.555555556$$

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.

$$5x-2y=12,2x+y=13$$

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

$$5x-2y=12$$

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

$$5x=2y+12$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+12\right)$$

Whakareatia \frac{1}{5} ki te 12+2y.

$$x=\frac{2}{5}y+\frac{12}{5}$$

Whakakapia te \frac{12+2y}{5} mō te x ki tērā atu whārite, 2x+y=13.

$$2\left(\frac{2}{5}y+\frac{12}{5}\right)+y=13$$

Whakareatia 2 ki te \frac{12+2y}{5}.

$$\frac{4}{5}y+\frac{24}{5}+y=13$$

Tāpiri \frac{4y}{5} ki te y.

$$\frac{9}{5}y+\frac{24}{5}=13$$

Me tango \frac{24}{5} mai i ngā taha e rua o te whārite.

$$\frac{9}{5}y=\frac{41}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{41}{9}$$

Whakaurua te \frac{41}{9} mō y ki x=\frac{2}{5}y+\frac{12}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times \frac{41}{9}+\frac{12}{5}$$

Whakareatia \frac{2}{5} ki te \frac{41}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{82}{45}+\frac{12}{5}$$

Tāpiri \frac{12}{5} ki te \frac{82}{45} 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.

$$x=\frac{38}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{38}{9},y=\frac{41}{9}$$

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.

$$5x-2y=12,2x+y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\2&1\end{matrix}\right))\left(\begin{matrix}5&-2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&1\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&1\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&1\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5-\left(-2\times 2\right)}&-\frac{-2}{5-\left(-2\times 2\right)}\\-\frac{2}{5-\left(-2\times 2\right)}&\frac{5}{5-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}12\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&\frac{2}{9}\\-\frac{2}{9}&\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}12\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 12+\frac{2}{9}\times 13\\-\frac{2}{9}\times 12+\frac{5}{9}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{38}{9}\\\frac{41}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{38}{9},y=\frac{41}{9}$$

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.

$$5x-2y=12,2x+y=13$$

Kia ōrite ai a 5x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$2\times 5x+2\left(-2\right)y=2\times 12,5\times 2x+5y=5\times 13$$

Whakarūnātia.

$$10x-4y=24,10x+5y=65$$

Me tango 10x+5y=65 mai i 10x-4y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-4y-5y=24-65$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-5y=24-65$$

Tāpiri -4y ki te -5y.

$$-9y=24-65$$

Tāpiri 24 ki te -65.

$$-9y=-41$$

Whakawehea ngā taha e rua ki te -9.

$$y=\frac{41}{9}$$

Whakaurua te \frac{41}{9} mō y ki 2x+y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+\frac{41}{9}=13$$

Me tango \frac{41}{9} mai i ngā taha e rua o te whārite.

$$2x=\frac{76}{9}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{38}{9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{38}{9},y=\frac{41}{9}$$