Whakaoti i te {l}{3x+2y=7}{4x+6y=13}

$$\left. \begin{array} { l } { 3 x + 2 y = 7 } \\ { 4 x + 6 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

$$y = \frac{11}{10} = 1\frac{1}{10} = 1.1$$

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.

$$3x+2y=7,4x+6y=13$$

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

$$3x+2y=7$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+7$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+7\right)$$

Whakareatia \frac{1}{3} ki te -2y+7.

$$x=-\frac{2}{3}y+\frac{7}{3}$$

Whakakapia te \frac{-2y+7}{3} mō te x ki tērā atu whārite, 4x+6y=13.

$$4\left(-\frac{2}{3}y+\frac{7}{3}\right)+6y=13$$

Whakareatia 4 ki te \frac{-2y+7}{3}.

$$-\frac{8}{3}y+\frac{28}{3}+6y=13$$

Tāpiri -\frac{8y}{3} ki te 6y.

$$\frac{10}{3}y+\frac{28}{3}=13$$

Me tango \frac{28}{3} mai i ngā taha e rua o te whārite.

$$\frac{10}{3}y=\frac{11}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{10}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{11}{10}$$

Whakaurua te \frac{11}{10} mō y ki x=-\frac{2}{3}y+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{11}{10}+\frac{7}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{11}{10} 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.

$$x=-\frac{11}{15}+\frac{7}{3}$$

Tāpiri \frac{7}{3} ki te -\frac{11}{15} 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.

$$x=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{5},y=\frac{11}{10}$$

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.

$$3x+2y=7,4x+6y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\4&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\4&6\end{matrix}\right))\left(\begin{matrix}3&2\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&6\end{matrix}\right))\left(\begin{matrix}7\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&6\end{matrix}\right))\left(\begin{matrix}7\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&6\end{matrix}\right))\left(\begin{matrix}7\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{3\times 6-2\times 4}&-\frac{2}{3\times 6-2\times 4}\\-\frac{4}{3\times 6-2\times 4}&\frac{3}{3\times 6-2\times 4}\end{matrix}\right)\left(\begin{matrix}7\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&-\frac{1}{5}\\-\frac{2}{5}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}7\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 7-\frac{1}{5}\times 13\\-\frac{2}{5}\times 7+\frac{3}{10}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5}\\\frac{11}{10}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{5},y=\frac{11}{10}$$

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.

$$3x+2y=7,4x+6y=13$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\times 2y=4\times 7,3\times 4x+3\times 6y=3\times 13$$

Whakarūnātia.

$$12x+8y=28,12x+18y=39$$

Me tango 12x+18y=39 mai i 12x+8y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+8y-18y=28-39$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-18y=28-39$$

Tāpiri 8y ki te -18y.

$$-10y=28-39$$

Tāpiri 28 ki te -39.

$$-10y=-11$$

Whakawehea ngā taha e rua ki te -10.

$$y=\frac{11}{10}$$

Whakaurua te \frac{11}{10} mō y ki 4x+6y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+6\times \frac{11}{10}=13$$

Whakareatia 6 ki te \frac{11}{10}.

$$4x+\frac{33}{5}=13$$

Me tango \frac{33}{5} mai i ngā taha e rua o te whārite.

$$4x=\frac{32}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{5},y=\frac{11}{10}$$