Whakaoti i te {l}{16x-10y=10}{-8x-6y=6}

$$\left. \begin{array} { l } { 16 x - 10 y = 10 } \\ { - 8 x - 6 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-1$$

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.

$$16x-10y=10,-8x-6y=6$$

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

$$16x-10y=10$$

Me tāpiri 10y ki ngā taha e rua o te whārite.

$$16x=10y+10$$

Whakawehea ngā taha e rua ki te 16.

$$x=\frac{1}{16}\left(10y+10\right)$$

Whakareatia \frac{1}{16} ki te 10+10y.

$$x=\frac{5}{8}y+\frac{5}{8}$$

Whakakapia te \frac{5+5y}{8} mō te x ki tērā atu whārite, -8x-6y=6.

$$-8\left(\frac{5}{8}y+\frac{5}{8}\right)-6y=6$$

Whakareatia -8 ki te \frac{5+5y}{8}.

$$-5y-5-6y=6$$

Tāpiri -5y ki te -6y.

$$-11y-5=6$$

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

$$-11y=11$$

Whakawehea ngā taha e rua ki te -11.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{5}{8}y+\frac{5}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{8}\left(-1\right)+\frac{5}{8}$$

Whakareatia \frac{5}{8} ki te -1.

$$x=\frac{-5+5}{8}$$

Tāpiri \frac{5}{8} ki te -\frac{5}{8} 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.

$$x=0$$

Kua oti te pūnaha te whakatau.

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

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.

$$16x-10y=10,-8x-6y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right))\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right))\left(\begin{matrix}10\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right))\left(\begin{matrix}10\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-10\\-8&-6\end{matrix}\right))\left(\begin{matrix}10\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{16\left(-6\right)-\left(-10\left(-8\right)\right)}&-\frac{-10}{16\left(-6\right)-\left(-10\left(-8\right)\right)}\\-\frac{-8}{16\left(-6\right)-\left(-10\left(-8\right)\right)}&\frac{16}{16\left(-6\right)-\left(-10\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}10\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{88}&-\frac{5}{88}\\-\frac{1}{22}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}10\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{88}\times 10-\frac{5}{88}\times 6\\-\frac{1}{22}\times 10-\frac{1}{11}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$16x-10y=10,-8x-6y=6$$

Kia ōrite ai a 16x me -8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 16.

$$-8\times 16x-8\left(-10\right)y=-8\times 10,16\left(-8\right)x+16\left(-6\right)y=16\times 6$$

Whakarūnātia.

$$-128x+80y=-80,-128x-96y=96$$

Me tango -128x-96y=96 mai i -128x+80y=-80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-128x+128x+80y+96y=-80-96$$

Tāpiri -128x ki te 128x. Ka whakakore atu ngā kupu -128x me 128x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$80y+96y=-80-96$$

Tāpiri 80y ki te 96y.

$$176y=-80-96$$

Tāpiri -80 ki te -96.

$$176y=-176$$

Whakawehea ngā taha e rua ki te 176.

$$y=-1$$

Whakaurua te -1 mō y ki -8x-6y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x-6\left(-1\right)=6$$

Whakareatia -6 ki te -1.

$$-8x+6=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-8x=0$$

Whakawehea ngā taha e rua ki te -8.

$$x=0$$

Kua oti te pūnaha te whakatau.

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