Whakaoti i te {l}{2x-3y+13=0}{3x-2y+12=0}

$$\left. \begin{array} { l } { 2 x - 3 y + 13 = 0 } \\ { 3 x - 2 y + 12 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=3$$

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.

$$2x-3y+13=0,3x-2y+12=0$$

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

$$2x-3y+13=0$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$2x-3y=-13$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$2x=3y-13$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y-13\right)$$

Whakareatia \frac{1}{2} ki te 3y-13.

$$x=\frac{3}{2}y-\frac{13}{2}$$

Whakakapia te \frac{3y-13}{2} mō te x ki tērā atu whārite, 3x-2y+12=0.

$$3\left(\frac{3}{2}y-\frac{13}{2}\right)-2y+12=0$$

Whakareatia 3 ki te \frac{3y-13}{2}.

$$\frac{9}{2}y-\frac{39}{2}-2y+12=0$$

Tāpiri \frac{9y}{2} ki te -2y.

$$\frac{5}{2}y-\frac{39}{2}+12=0$$

Tāpiri -\frac{39}{2} ki te 12.

$$\frac{5}{2}y-\frac{15}{2}=0$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=\frac{15}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{3}{2}y-\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times 3-\frac{13}{2}$$

Whakareatia \frac{3}{2} ki te 3.

$$x=\frac{9-13}{2}$$

Tāpiri -\frac{13}{2} ki te \frac{9}{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.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

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.

$$2x-3y+13=0,3x-2y+12=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-\left(-3\times 3\right)}&-\frac{-3}{2\left(-2\right)-\left(-3\times 3\right)}\\-\frac{3}{2\left(-2\right)-\left(-3\times 3\right)}&\frac{2}{2\left(-2\right)-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{3}{5}\\-\frac{3}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-13\\-12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\left(-13\right)+\frac{3}{5}\left(-12\right)\\-\frac{3}{5}\left(-13\right)+\frac{2}{5}\left(-12\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x-3y+13=0,3x-2y+12=0$$

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.

$$3\times 2x+3\left(-3\right)y+3\times 13=0,2\times 3x+2\left(-2\right)y+2\times 12=0$$

Whakarūnātia.

$$6x-9y+39=0,6x-4y+24=0$$

Me tango 6x-4y+24=0 mai i 6x-9y+39=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-9y+4y+39-24=0$$

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.

$$-9y+4y+39-24=0$$

Tāpiri -9y ki te 4y.

$$-5y+39-24=0$$

Tāpiri 39 ki te -24.

$$-5y+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

Whakaurua te 3 mō y ki 3x-2y+12=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times 3+12=0$$

Whakareatia -2 ki te 3.

$$3x-6+12=0$$

Tāpiri -6 ki te 12.

$$3x+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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