Whakaoti i te {c}{8x+6y=-10}{-8x-5y=15}

$$\left. \begin{array} { c } { 8 x + 6 y = - 10 } \\ { - 8 x - 5 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=5$$

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.

$$8x+6y=-10,-8x-5y=15$$

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

$$8x+6y=-10$$

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

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

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-6y-10\right)$$

Whakareatia \frac{1}{8} ki te -6y-10.

$$x=-\frac{3}{4}y-\frac{5}{4}$$

Whakakapia te \frac{-3y-5}{4} mō te x ki tērā atu whārite, -8x-5y=15.

$$-8\left(-\frac{3}{4}y-\frac{5}{4}\right)-5y=15$$

Whakareatia -8 ki te \frac{-3y-5}{4}.

$$6y+10-5y=15$$

Tāpiri 6y ki te -5y.

$$y+10=15$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{3}{4}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.

$$x=-\frac{3}{4}\times 5-\frac{5}{4}$$

Whakareatia -\frac{3}{4} ki te 5.

$$x=\frac{-15-5}{4}$$

Tāpiri -\frac{5}{4} ki te -\frac{15}{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.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=5$$

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.

$$8x+6y=-10,-8x-5y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&6\\-8&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right))\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right))\left(\begin{matrix}-10\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right))\left(\begin{matrix}-10\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&6\\-8&-5\end{matrix}\right))\left(\begin{matrix}-10\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8\left(-5\right)-6\left(-8\right)}&-\frac{6}{8\left(-5\right)-6\left(-8\right)}\\-\frac{-8}{8\left(-5\right)-6\left(-8\right)}&\frac{8}{8\left(-5\right)-6\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-10\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8}&-\frac{3}{4}\\1&1\end{matrix}\right)\left(\begin{matrix}-10\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8}\left(-10\right)-\frac{3}{4}\times 15\\-10+15\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=5$$

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.

$$8x+6y=-10,-8x-5y=15$$

Kia ōrite ai a 8x 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 8.

$$-8\times 8x-8\times 6y=-8\left(-10\right),8\left(-8\right)x+8\left(-5\right)y=8\times 15$$

Whakarūnātia.

$$-64x-48y=80,-64x-40y=120$$

Me tango -64x-40y=120 mai i -64x-48y=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-64x+64x-48y+40y=80-120$$

Tāpiri -64x ki te 64x. Ka whakakore atu ngā kupu -64x me 64x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-48y+40y=80-120$$

Tāpiri -48y ki te 40y.

$$-8y=80-120$$

Tāpiri 80 ki te -120.

$$-8y=-40$$

Whakawehea ngā taha e rua ki te -8.

$$y=5$$

Whakaurua te 5 mō y ki -8x-5y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x-5\times 5=15$$

Whakareatia -5 ki te 5.

$$-8x-25=15$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$-8x=40$$

Whakawehea ngā taha e rua ki te -8.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=5$$