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

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

Whakaoti mō y, x

$$x=-1$$

$$y=0$$

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.

$$-10y+9x=-9,10y+5x=-5$$

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

$$-10y+9x=-9$$

Me tango 9x mai i ngā taha e rua o te whārite.

$$-10y=-9x-9$$

Whakawehea ngā taha e rua ki te -10.

$$y=-\frac{1}{10}\left(-9x-9\right)$$

Whakareatia -\frac{1}{10} ki te -9x-9.

$$y=\frac{9}{10}x+\frac{9}{10}$$

Whakakapia te \frac{9+9x}{10} mō te y ki tērā atu whārite, 10y+5x=-5.

$$10\left(\frac{9}{10}x+\frac{9}{10}\right)+5x=-5$$

Whakareatia 10 ki te \frac{9+9x}{10}.

$$9x+9+5x=-5$$

Tāpiri 9x ki te 5x.

$$14x+9=-5$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$14x=-14$$

Whakawehea ngā taha e rua ki te 14.

$$x=-1$$

Whakaurua te -1 mō x ki y=\frac{9}{10}x+\frac{9}{10}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{9}{10}\left(-1\right)+\frac{9}{10}$$

Whakareatia \frac{9}{10} ki te -1.

$$y=\frac{-9+9}{10}$$

Tāpiri \frac{9}{10} ki te -\frac{9}{10} 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.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$y=0,x=-1$$

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.

$$-10y+9x=-9,10y+5x=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{28}&\frac{9}{140}\\\frac{1}{14}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}-9\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{28}\left(-9\right)+\frac{9}{140}\left(-5\right)\\\frac{1}{14}\left(-9\right)+\frac{1}{14}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=0,x=-1$$

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.

$$-10y+9x=-9,10y+5x=-5$$

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

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

Whakarūnātia.

$$-100y+90x=-90,-100y-50x=50$$

Me tango -100y-50x=50 mai i -100y+90x=-90 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-100y+100y+90x+50x=-90-50$$

Tāpiri -100y ki te 100y. Ka whakakore atu ngā kupu -100y me 100y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$90x+50x=-90-50$$

Tāpiri 90x ki te 50x.

$$140x=-90-50$$

Tāpiri -90 ki te -50.

$$140x=-140$$

Whakawehea ngā taha e rua ki te 140.

$$x=-1$$

Whakaurua te -1 mō x ki 10y+5x=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$10y+5\left(-1\right)=-5$$

Whakareatia 5 ki te -1.

$$10y-5=-5$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$10y=0$$

Whakawehea ngā taha e rua ki te 10.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$y=0,x=-1$$