Whakaoti i te {l}{6x+3y=24}{7x+6y=33}

$$\left. \begin{array} { l } { 6 x + 3 y = 24 } \\ { 7 x + 6 y = 33 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=2$$

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.

$$6x+3y=24,7x+6y=33$$

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

$$6x+3y=24$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$6x=-3y+24$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-3y+24\right)$$

Whakareatia \frac{1}{6} ki te -3y+24.

$$x=-\frac{1}{2}y+4$$

Whakakapia te -\frac{y}{2}+4 mō te x ki tērā atu whārite, 7x+6y=33.

$$7\left(-\frac{1}{2}y+4\right)+6y=33$$

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

$$-\frac{7}{2}y+28+6y=33$$

Tāpiri -\frac{7y}{2} ki te 6y.

$$\frac{5}{2}y+28=33$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$\frac{5}{2}y=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{1}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 2+4$$

Whakareatia -\frac{1}{2} ki te 2.

$$x=-1+4$$

Tāpiri 4 ki te -1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$

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.

$$6x+3y=24,7x+6y=33$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&3\\7&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\33\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&3\\7&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&3\\7&6\end{matrix}\right))\left(\begin{matrix}6&3\\7&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\7&6\end{matrix}\right))\left(\begin{matrix}24\\33\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\7&6\end{matrix}\right))\left(\begin{matrix}24\\33\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\7&6\end{matrix}\right))\left(\begin{matrix}24\\33\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{6\times 6-3\times 7}&-\frac{3}{6\times 6-3\times 7}\\-\frac{7}{6\times 6-3\times 7}&\frac{6}{6\times 6-3\times 7}\end{matrix}\right)\left(\begin{matrix}24\\33\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\-\frac{7}{15}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}24\\33\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 24-\frac{1}{5}\times 33\\-\frac{7}{15}\times 24+\frac{2}{5}\times 33\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=2$$

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.

$$6x+3y=24,7x+6y=33$$

Kia ōrite ai a 6x 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 6.

$$7\times 6x+7\times 3y=7\times 24,6\times 7x+6\times 6y=6\times 33$$

Whakarūnātia.

$$42x+21y=168,42x+36y=198$$

Me tango 42x+36y=198 mai i 42x+21y=168 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$42x-42x+21y-36y=168-198$$

Tāpiri 42x ki te -42x. Ka whakakore atu ngā kupu 42x me -42x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y-36y=168-198$$

Tāpiri 21y ki te -36y.

$$-15y=168-198$$

Tāpiri 168 ki te -198.

$$-15y=-30$$

Whakawehea ngā taha e rua ki te -15.

$$y=2$$

Whakaurua te 2 mō y ki 7x+6y=33. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+6\times 2=33$$

Whakareatia 6 ki te 2.

$$7x+12=33$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$7x=21$$

Whakawehea ngā taha e rua ki te 7.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$