Whakaoti i te {l}{3x+4y=3}{8x+7y=14}

$$\left. \begin{array} { l } { 3 x + 4 y = 3 } \\ { 8 x + 7 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{35}{11} = 3\frac{2}{11} \approx 3.181818182$$

$$y = -\frac{18}{11} = -1\frac{7}{11} \approx -1.636363636$$

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.

$$3x+4y=3,8x+7y=14$$

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

$$3x+4y=3$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$3x=-4y+3$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+3\right)$$

Whakareatia \frac{1}{3} ki te -4y+3.

$$x=-\frac{4}{3}y+1$$

Whakakapia te -\frac{4y}{3}+1 mō te x ki tērā atu whārite, 8x+7y=14.

$$8\left(-\frac{4}{3}y+1\right)+7y=14$$

Whakareatia 8 ki te -\frac{4y}{3}+1.

$$-\frac{32}{3}y+8+7y=14$$

Tāpiri -\frac{32y}{3} ki te 7y.

$$-\frac{11}{3}y+8=14$$

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

$$-\frac{11}{3}y=6$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{18}{11}$$

Whakaurua te -\frac{18}{11} mō y ki x=-\frac{4}{3}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-\frac{18}{11}\right)+1$$

Whakareatia -\frac{4}{3} ki te -\frac{18}{11} 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.

$$x=\frac{24}{11}+1$$

Tāpiri 1 ki te \frac{24}{11}.

$$x=\frac{35}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{11},y=-\frac{18}{11}$$

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.

$$3x+4y=3,8x+7y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\8&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\8&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\8&7\end{matrix}\right))\left(\begin{matrix}3&4\\8&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\8&7\end{matrix}\right))\left(\begin{matrix}3\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\8&7\end{matrix}\right))\left(\begin{matrix}3\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\8&7\end{matrix}\right))\left(\begin{matrix}3\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3\times 7-4\times 8}&-\frac{4}{3\times 7-4\times 8}\\-\frac{8}{3\times 7-4\times 8}&\frac{3}{3\times 7-4\times 8}\end{matrix}\right)\left(\begin{matrix}3\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{11}&\frac{4}{11}\\\frac{8}{11}&-\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}3\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{11}\times 3+\frac{4}{11}\times 14\\\frac{8}{11}\times 3-\frac{3}{11}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{11}\\-\frac{18}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{35}{11},y=-\frac{18}{11}$$

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.

$$3x+4y=3,8x+7y=14$$

Kia ōrite ai a 3x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$8\times 3x+8\times 4y=8\times 3,3\times 8x+3\times 7y=3\times 14$$

Whakarūnātia.

$$24x+32y=24,24x+21y=42$$

Me tango 24x+21y=42 mai i 24x+32y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x+32y-21y=24-42$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$32y-21y=24-42$$

Tāpiri 32y ki te -21y.

$$11y=24-42$$

Tāpiri 24 ki te -42.

$$11y=-18$$

Whakawehea ngā taha e rua ki te 11.

$$y=-\frac{18}{11}$$

Whakaurua te -\frac{18}{11} mō y ki 8x+7y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+7\left(-\frac{18}{11}\right)=14$$

Whakareatia 7 ki te -\frac{18}{11}.

$$8x-\frac{126}{11}=14$$

Me tāpiri \frac{126}{11} ki ngā taha e rua o te whārite.

$$8x=\frac{280}{11}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{35}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{11},y=-\frac{18}{11}$$