Whakaoti i te {l}{3x+4y=1}{2x+3y=-1}

$$\left. \begin{array} { l } { 3 x + 4 y = 1 } \\ { 2 x + 3 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=-5$$

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.

$$3x+4y=1,2x+3y=-1$$

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

$$3x+4y=1$$

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

$$3x=-4y+1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+1\right)$$

Whakareatia \frac{1}{3} ki te -4y+1.

$$x=-\frac{4}{3}y+\frac{1}{3}$$

Whakakapia te \frac{-4y+1}{3} mō te x ki tērā atu whārite, 2x+3y=-1.

$$2\left(-\frac{4}{3}y+\frac{1}{3}\right)+3y=-1$$

Whakareatia 2 ki te \frac{-4y+1}{3}.

$$-\frac{8}{3}y+\frac{2}{3}+3y=-1$$

Tāpiri -\frac{8y}{3} ki te 3y.

$$\frac{1}{3}y+\frac{2}{3}=-1$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}y=-\frac{5}{3}$$

Me whakarea ngā taha e rua ki te 3.

$$y=-5$$

Whakaurua te -5 mō y ki x=-\frac{4}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-5\right)+\frac{1}{3}$$

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

$$x=\frac{20+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{20}{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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=-5$$

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.

$$3x+4y=1,2x+3y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\2&3\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\-5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=-5$$

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.

$$3x+4y=1,2x+3y=-1$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\times 4y=2,3\times 2x+3\times 3y=3\left(-1\right)$$

Whakarūnātia.

$$6x+8y=2,6x+9y=-3$$

Me tango 6x+9y=-3 mai i 6x+8y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+8y-9y=2+3$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-9y=2+3$$

Tāpiri 8y ki te -9y.

$$-y=2+3$$

Tāpiri 2 ki te 3.

$$-y=5$$

Whakawehea ngā taha e rua ki te -1.

$$y=-5$$

Whakaurua te -5 mō y ki 2x+3y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-5\right)=-1$$

Whakareatia 3 ki te -5.

$$2x-15=-1$$

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

$$2x=14$$

Whakawehea ngā taha e rua ki te 2.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=-5$$