Whakaoti i te {l}{-8x-9y=-10}{-4x-3y=10}

$$\left. \begin{array} { l } { - 8 x - 9 y = - 10 } \\ { - 4 x - 3 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-10$$

$$y=10$$

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.

$$-8x-9y=-10,-4x-3y=10$$

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

$$-8x-9y=-10$$

Me tāpiri 9y ki ngā taha e rua o te whārite.

$$-8x=9y-10$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(9y-10\right)$$

Whakareatia -\frac{1}{8} ki te 9y-10.

$$x=-\frac{9}{8}y+\frac{5}{4}$$

Whakakapia te -\frac{9y}{8}+\frac{5}{4} mō te x ki tērā atu whārite, -4x-3y=10.

$$-4\left(-\frac{9}{8}y+\frac{5}{4}\right)-3y=10$$

Whakareatia -4 ki te -\frac{9y}{8}+\frac{5}{4}.

$$\frac{9}{2}y-5-3y=10$$

Tāpiri \frac{9y}{2} ki te -3y.

$$\frac{3}{2}y-5=10$$

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

$$\frac{3}{2}y=15$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki x=-\frac{9}{8}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{8}\times 10+\frac{5}{4}$$

Whakareatia -\frac{9}{8} ki te 10.

$$x=\frac{-45+5}{4}$$

Tāpiri \frac{5}{4} ki te -\frac{45}{4} 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.

$$x=-10$$

Kua oti te pūnaha te whakatau.

$$x=-10,y=10$$

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.

$$-8x-9y=-10,-4x-3y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right))\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-9\\-4&-3\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-8\left(-3\right)-\left(-9\left(-4\right)\right)}&-\frac{-9}{-8\left(-3\right)-\left(-9\left(-4\right)\right)}\\-\frac{-4}{-8\left(-3\right)-\left(-9\left(-4\right)\right)}&-\frac{8}{-8\left(-3\right)-\left(-9\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}-10\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{3}{4}\\-\frac{1}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}-10\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-10\right)-\frac{3}{4}\times 10\\-\frac{1}{3}\left(-10\right)+\frac{2}{3}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-10,y=10$$

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.

$$-8x-9y=-10,-4x-3y=10$$

Kia ōrite ai a -8x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -8.

$$-4\left(-8\right)x-4\left(-9\right)y=-4\left(-10\right),-8\left(-4\right)x-8\left(-3\right)y=-8\times 10$$

Whakarūnātia.

$$32x+36y=40,32x+24y=-80$$

Me tango 32x+24y=-80 mai i 32x+36y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32x-32x+36y-24y=40+80$$

Tāpiri 32x ki te -32x. Ka whakakore atu ngā kupu 32x me -32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y-24y=40+80$$

Tāpiri 36y ki te -24y.

$$12y=40+80$$

Tāpiri 40 ki te 80.

$$12y=120$$

Whakawehea ngā taha e rua ki te 12.

$$y=10$$

Whakaurua te 10 mō y ki -4x-3y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x-3\times 10=10$$

Whakareatia -3 ki te 10.

$$-4x-30=10$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$-4x=40$$

Whakawehea ngā taha e rua ki te -4.

$$x=-10$$

Kua oti te pūnaha te whakatau.

$$x=-10,y=10$$