Whakaoti i te {c}{3x-y=8}{x+y=4}

$$\left\{ \begin{array} { c } { 3 x - y = 8 } \\ { x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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.

$$3x-y=8,x+y=4$$

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

$$3x-y=8$$

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

$$3x=y+8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+8\right)$$

Whakareatia \frac{1}{3} ki te y+8.

$$x=\frac{1}{3}y+\frac{8}{3}$$

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

$$\frac{1}{3}y+\frac{8}{3}+y=4$$

Tāpiri \frac{y}{3} ki te y.

$$\frac{4}{3}y+\frac{8}{3}=4$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$\frac{4}{3}y=\frac{4}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{1}{3}y+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1+8}{3}$$

Tāpiri \frac{8}{3} ki te \frac{1}{3} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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.

$$3x-y=8,x+y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 8+\frac{1}{4}\times 4\\-\frac{1}{4}\times 8+\frac{3}{4}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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.

$$3x-y=8,x+y=4$$

Kia ōrite ai a 3x 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 3.

$$3x-y=8,3x+3y=3\times 4$$

Whakarūnātia.

$$3x-y=8,3x+3y=12$$

Me tango 3x+3y=12 mai i 3x-y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-y-3y=8-12$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-y-3y=8-12$$

Tāpiri -y ki te -3y.

$$-4y=8-12$$

Tāpiri 8 ki te -12.

$$-4y=-4$$

Whakawehea ngā taha e rua ki te -4.

$$y=1$$

Whakaurua te 1 mō y ki x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+1=4$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$