Whakaoti i te {l}{-12x+10y=-10}{6x-7y=-5}

$$\left. \begin{array} { l } { - 12 x + 10 y = - 10 } \\ { 6 x - 7 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=5$$

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.

$$-12x+10y=-10,6x-7y=-5$$

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

$$-12x+10y=-10$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$-12x=-10y-10$$

Whakawehea ngā taha e rua ki te -12.

$$x=-\frac{1}{12}\left(-10y-10\right)$$

Whakareatia -\frac{1}{12} ki te -10y-10.

$$x=\frac{5}{6}y+\frac{5}{6}$$

Whakakapia te \frac{5+5y}{6} mō te x ki tērā atu whārite, 6x-7y=-5.

$$6\left(\frac{5}{6}y+\frac{5}{6}\right)-7y=-5$$

Whakareatia 6 ki te \frac{5+5y}{6}.

$$5y+5-7y=-5$$

Tāpiri 5y ki te -7y.

$$-2y+5=-5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-2y=-10$$

Whakawehea ngā taha e rua ki te -2.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{5}{6}y+\frac{5}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{6}\times 5+\frac{5}{6}$$

Whakareatia \frac{5}{6} ki te 5.

$$x=\frac{25+5}{6}$$

Tāpiri \frac{5}{6} ki te \frac{25}{6} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=5$$

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.

$$-12x+10y=-10,6x-7y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-12&10\\6&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right))\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right))\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right))\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&10\\6&-7\end{matrix}\right))\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-12\left(-7\right)-10\times 6}&-\frac{10}{-12\left(-7\right)-10\times 6}\\-\frac{6}{-12\left(-7\right)-10\times 6}&-\frac{12}{-12\left(-7\right)-10\times 6}\end{matrix}\right)\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{24}&-\frac{5}{12}\\-\frac{1}{4}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-10\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{24}\left(-10\right)-\frac{5}{12}\left(-5\right)\\-\frac{1}{4}\left(-10\right)-\frac{1}{2}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=5$$

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.

$$-12x+10y=-10,6x-7y=-5$$

Kia ōrite ai a -12x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -12.

$$6\left(-12\right)x+6\times 10y=6\left(-10\right),-12\times 6x-12\left(-7\right)y=-12\left(-5\right)$$

Whakarūnātia.

$$-72x+60y=-60,-72x+84y=60$$

Me tango -72x+84y=60 mai i -72x+60y=-60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-72x+72x+60y-84y=-60-60$$

Tāpiri -72x ki te 72x. Ka whakakore atu ngā kupu -72x me 72x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$60y-84y=-60-60$$

Tāpiri 60y ki te -84y.

$$-24y=-60-60$$

Tāpiri -60 ki te -60.

$$-24y=-120$$

Whakawehea ngā taha e rua ki te -24.

$$y=5$$

Whakaurua te 5 mō y ki 6x-7y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-7\times 5=-5$$

Whakareatia -7 ki te 5.

$$6x-35=-5$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$6x=30$$

Whakawehea ngā taha e rua ki te 6.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=5$$