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

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

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

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

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

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

Me tango 3y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

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

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

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

$$\frac{9}{2}y-\frac{3}{2}-4y=-1$$

Tāpiri \frac{9y}{2} ki te -4y.

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

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$\frac{1}{2}y=\frac{1}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$y=1$$

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

$$x=\frac{3-1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{3}{2} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

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

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

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

Whakarūnātia.

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

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

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

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.

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

Tāpiri 9y ki te -8y.

$$y=3-2$$

Tāpiri 3 ki te -2.

$$y=1$$

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

$$3x-4=-1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$