Whakaoti i te {l}{x+4y=2}{-x-3y=3}

$$\left. \begin{array} { l } { x + 4 y = 2 } \\ { - x - 3 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-18$$

$$y=5$$

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.

$$x+4y=2,-x-3y=3$$

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

$$x+4y=2$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$x=-4y+2$$

Whakakapia te -4y+2 mō te x ki tērā atu whārite, -x-3y=3.

$$-\left(-4y+2\right)-3y=3$$

Whakareatia -1 ki te -4y+2.

$$4y-2-3y=3$$

Tāpiri 4y ki te -3y.

$$y-2=3$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$y=5$$

Whakaurua te 5 mō y ki x=-4y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times 5+2$$

Whakareatia -4 ki te 5.

$$x=-20+2$$

Tāpiri 2 ki te -20.

$$x=-18$$

Kua oti te pūnaha te whakatau.

$$x=-18,y=5$$

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.

$$x+4y=2,-x-3y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\-1&-3\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-1&-3\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-18,y=5$$

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.

$$x+4y=2,-x-3y=3$$

Kia ōrite ai a x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-x-4y=-2,-x-3y=3$$

Me tango -x-3y=3 mai i -x-4y=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-4y+3y=-2-3$$

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

$$-4y+3y=-2-3$$

Tāpiri -4y ki te 3y.

$$-y=-2-3$$

Tāpiri -2 ki te -3.

$$-y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$y=5$$

Whakaurua te 5 mō y ki -x-3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x-3\times 5=3$$

Whakareatia -3 ki te 5.

$$-x-15=3$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$-x=18$$

Whakawehea ngā taha e rua ki te -1.

$$x=-18$$

Kua oti te pūnaha te whakatau.

$$x=-18,y=5$$