Whakaoti i te {l}{-4x-10y=20}{8x+10y=20}

$$\left. \begin{array} { l } { - 4 x - 10 y = 20 } \\ { 8 x + 10 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-6$$

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.

$$-4x-10y=20,8x+10y=20$$

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

$$-4x-10y=20$$

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

$$-4x=10y+20$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(10y+20\right)$$

Whakareatia -\frac{1}{4} ki te 20+10y.

$$x=-\frac{5}{2}y-5$$

Whakakapia te -\frac{5y}{2}-5 mō te x ki tērā atu whārite, 8x+10y=20.

$$8\left(-\frac{5}{2}y-5\right)+10y=20$$

Whakareatia 8 ki te -\frac{5y}{2}-5.

$$-20y-40+10y=20$$

Tāpiri -20y ki te 10y.

$$-10y-40=20$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$-10y=60$$

Whakawehea ngā taha e rua ki te -10.

$$y=-6$$

Whakaurua te -6 mō y ki x=-\frac{5}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\left(-6\right)-5$$

Whakareatia -\frac{5}{2} ki te -6.

$$x=15-5$$

Tāpiri -5 ki te 15.

$$x=10$$

Kua oti te pūnaha te whakatau.

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

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.

$$-4x-10y=20,8x+10y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&-10\\8&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right))\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-10\\8&10\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{-4\times 10-\left(-10\times 8\right)}&-\frac{-10}{-4\times 10-\left(-10\times 8\right)}\\-\frac{8}{-4\times 10-\left(-10\times 8\right)}&-\frac{4}{-4\times 10-\left(-10\times 8\right)}\end{matrix}\right)\left(\begin{matrix}20\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{5}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}20\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 20+\frac{1}{4}\times 20\\-\frac{1}{5}\times 20-\frac{1}{10}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$-4x-10y=20,8x+10y=20$$

Kia ōrite ai a -4x 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 -4.

$$8\left(-4\right)x+8\left(-10\right)y=8\times 20,-4\times 8x-4\times 10y=-4\times 20$$

Whakarūnātia.

$$-32x-80y=160,-32x-40y=-80$$

Me tango -32x-40y=-80 mai i -32x-80y=160 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-32x+32x-80y+40y=160+80$$

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.

$$-80y+40y=160+80$$

Tāpiri -80y ki te 40y.

$$-40y=160+80$$

Tāpiri 160 ki te 80.

$$-40y=240$$

Whakawehea ngā taha e rua ki te -40.

$$y=-6$$

Whakaurua te -6 mō y ki 8x+10y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+10\left(-6\right)=20$$

Whakareatia 10 ki te -6.

$$8x-60=20$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$8x=80$$

Whakawehea ngā taha e rua ki te 8.

$$x=10$$

Kua oti te pūnaha te whakatau.

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