Whakaoti i te {l}{-10x+2y=-8}{10x-y=9}

$$\left. \begin{array} { l } { - 10 x + 2 y = - 8 } \\ { 10 x - y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

$$-10x+2y=-8,10x-y=9$$

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

$$-10x+2y=-8$$

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

$$-10x=-2y-8$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(-2y-8\right)$$

Whakareatia -\frac{1}{10} ki te -2y-8.

$$x=\frac{1}{5}y+\frac{4}{5}$$

Whakakapia te \frac{4+y}{5} mō te x ki tērā atu whārite, 10x-y=9.

$$10\left(\frac{1}{5}y+\frac{4}{5}\right)-y=9$$

Whakareatia 10 ki te \frac{4+y}{5}.

$$2y+8-y=9$$

Tāpiri 2y ki te -y.

$$y+8=9$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{1}{5}y+\frac{4}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1+4}{5}$$

Tāpiri \frac{4}{5} ki te \frac{1}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

$$-10x+2y=-8,10x-y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&2\\10&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right))\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right))\left(\begin{matrix}-8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right))\left(\begin{matrix}-8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&2\\10&-1\end{matrix}\right))\left(\begin{matrix}-8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-10\left(-1\right)-2\times 10}&-\frac{2}{-10\left(-1\right)-2\times 10}\\-\frac{10}{-10\left(-1\right)-2\times 10}&-\frac{10}{-10\left(-1\right)-2\times 10}\end{matrix}\right)\left(\begin{matrix}-8\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{5}\\1&1\end{matrix}\right)\left(\begin{matrix}-8\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\left(-8\right)+\frac{1}{5}\times 9\\-8+9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

$$-10x+2y=-8,10x-y=9$$

Kia ōrite ai a -10x me 10x, 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.

$$10\left(-10\right)x+10\times 2y=10\left(-8\right),-10\times 10x-10\left(-1\right)y=-10\times 9$$

Whakarūnātia.

$$-100x+20y=-80,-100x+10y=-90$$

Me tango -100x+10y=-90 mai i -100x+20y=-80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-100x+100x+20y-10y=-80+90$$

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

$$20y-10y=-80+90$$

Tāpiri 20y ki te -10y.

$$10y=-80+90$$

Tāpiri -80 ki te 90.

$$10y=10$$

Whakawehea ngā taha e rua ki te 10.

$$y=1$$

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

$$10x-1=9$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$10x=10$$

Whakawehea ngā taha e rua ki te 10.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$