Whakaoti i te {l}{10x+4y=-12}{-9x-5y=1}

$$\left. \begin{array} { l } { 10 x + 4 y = - 12 } \\ { - 9 x - 5 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=7$$

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.

$$10x+4y=-12,-9x-5y=1$$

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

$$10x+4y=-12$$

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

$$10x=-4y-12$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-4y-12\right)$$

Whakareatia \frac{1}{10} ki te -4y-12.

$$x=-\frac{2}{5}y-\frac{6}{5}$$

Whakakapia te \frac{-2y-6}{5} mō te x ki tērā atu whārite, -9x-5y=1.

$$-9\left(-\frac{2}{5}y-\frac{6}{5}\right)-5y=1$$

Whakareatia -9 ki te \frac{-2y-6}{5}.

$$\frac{18}{5}y+\frac{54}{5}-5y=1$$

Tāpiri \frac{18y}{5} ki te -5y.

$$-\frac{7}{5}y+\frac{54}{5}=1$$

Me tango \frac{54}{5} mai i ngā taha e rua o te whārite.

$$-\frac{7}{5}y=-\frac{49}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{2}{5}y-\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\times 7-\frac{6}{5}$$

Whakareatia -\frac{2}{5} ki te 7.

$$x=\frac{-14-6}{5}$$

Tāpiri -\frac{6}{5} ki te -\frac{14}{5} 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.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=7$$

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.

$$10x+4y=-12,-9x-5y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&4\\-9&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right))\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right))\left(\begin{matrix}-12\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right))\left(\begin{matrix}-12\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&4\\-9&-5\end{matrix}\right))\left(\begin{matrix}-12\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{10\left(-5\right)-4\left(-9\right)}&-\frac{4}{10\left(-5\right)-4\left(-9\right)}\\-\frac{-9}{10\left(-5\right)-4\left(-9\right)}&\frac{10}{10\left(-5\right)-4\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-12\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}&\frac{2}{7}\\-\frac{9}{14}&-\frac{5}{7}\end{matrix}\right)\left(\begin{matrix}-12\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{14}\left(-12\right)+\frac{2}{7}\\-\frac{9}{14}\left(-12\right)-\frac{5}{7}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=7$$

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.

$$10x+4y=-12,-9x-5y=1$$

Kia ōrite ai a 10x me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 10.

$$-9\times 10x-9\times 4y=-9\left(-12\right),10\left(-9\right)x+10\left(-5\right)y=10$$

Whakarūnātia.

$$-90x-36y=108,-90x-50y=10$$

Me tango -90x-50y=10 mai i -90x-36y=108 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-90x+90x-36y+50y=108-10$$

Tāpiri -90x ki te 90x. Ka whakakore atu ngā kupu -90x me 90x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-36y+50y=108-10$$

Tāpiri -36y ki te 50y.

$$14y=108-10$$

Tāpiri 108 ki te -10.

$$14y=98$$

Whakawehea ngā taha e rua ki te 14.

$$y=7$$

Whakaurua te 7 mō y ki -9x-5y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x-5\times 7=1$$

Whakareatia -5 ki te 7.

$$-9x-35=1$$

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

$$-9x=36$$

Whakawehea ngā taha e rua ki te -9.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=7$$