Whakaoti i te {l}{2x+y=-16}{-4x+10y=8}

$$\left. \begin{array} { l } { 2 x + y = - 16 } \\ { - 4 x + 10 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=-2$$

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.

$$2x+y=-16,-4x+10y=8$$

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

$$2x+y=-16$$

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

$$2x=-y-16$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y-16\right)$$

Whakareatia \frac{1}{2} ki te -y-16.

$$x=-\frac{1}{2}y-8$$

Whakakapia te -\frac{y}{2}-8 mō te x ki tērā atu whārite, -4x+10y=8.

$$-4\left(-\frac{1}{2}y-8\right)+10y=8$$

Whakareatia -4 ki te -\frac{y}{2}-8.

$$2y+32+10y=8$$

Tāpiri 2y ki te 10y.

$$12y+32=8$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$12y=-24$$

Whakawehea ngā taha e rua ki te 12.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{1}{2}y-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia -\frac{1}{2} ki te -2.

$$x=1-8$$

Tāpiri -8 ki te 1.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-2$$

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.

$$2x+y=-16,-4x+10y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\-4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\-4&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\-4&10\end{matrix}\right))\left(\begin{matrix}2&1\\-4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&10\end{matrix}\right))\left(\begin{matrix}-16\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&10\end{matrix}\right))\left(\begin{matrix}-16\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&10\end{matrix}\right))\left(\begin{matrix}-16\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{2\times 10-\left(-4\right)}&-\frac{1}{2\times 10-\left(-4\right)}\\-\frac{-4}{2\times 10-\left(-4\right)}&\frac{2}{2\times 10-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-16\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}&-\frac{1}{24}\\\frac{1}{6}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-16\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}\left(-16\right)-\frac{1}{24}\times 8\\\frac{1}{6}\left(-16\right)+\frac{1}{12}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=-2$$

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.

$$2x+y=-16,-4x+10y=8$$

Kia ōrite ai a 2x 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 2.

$$-4\times 2x-4y=-4\left(-16\right),2\left(-4\right)x+2\times 10y=2\times 8$$

Whakarūnātia.

$$-8x-4y=64,-8x+20y=16$$

Me tango -8x+20y=16 mai i -8x-4y=64 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x-4y-20y=64-16$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-20y=64-16$$

Tāpiri -4y ki te -20y.

$$-24y=64-16$$

Tāpiri 64 ki te -16.

$$-24y=48$$

Whakawehea ngā taha e rua ki te -24.

$$y=-2$$

Whakaurua te -2 mō y ki -4x+10y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+10\left(-2\right)=8$$

Whakareatia 10 ki te -2.

$$-4x-20=8$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$-4x=28$$

Whakawehea ngā taha e rua ki te -4.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-2$$