Whakaoti i te {l}{2x+3y=4}{3x+2y=7}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 4 } \\ { 3 x + 2 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{13}{5} = 2\frac{3}{5} = 2.6$$

$$y=-\frac{2}{5}=-0.4$$

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.

$$2x+3y=4,3x+2y=7$$

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

$$2x+3y=4$$

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

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

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -\frac{3y}{2}+2 mō te x ki tērā atu whārite, 3x+2y=7.

$$3\left(-\frac{3}{2}y+2\right)+2y=7$$

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

$$-\frac{9}{2}y+6+2y=7$$

Tāpiri -\frac{9y}{2} ki te 2y.

$$-\frac{5}{2}y+6=7$$

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

$$-\frac{5}{2}y=1$$

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.

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

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

$$x=-\frac{3}{2}\left(-\frac{2}{5}\right)+2$$

Whakareatia -\frac{3}{2} ki te -\frac{2}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{3}{5}+2$$

Tāpiri 2 ki te \frac{3}{5}.

$$x=\frac{13}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{5},y=-\frac{2}{5}$$

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.

$$2x+3y=4,3x+2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\times 3}&-\frac{3}{2\times 2-3\times 3}\\-\frac{3}{2\times 2-3\times 3}&\frac{2}{2\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{3}{5}\\\frac{3}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\times 4+\frac{3}{5}\times 7\\\frac{3}{5}\times 4-\frac{2}{5}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{13}{5},y=-\frac{2}{5}$$

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.

$$2x+3y=4,3x+2y=7$$

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.

$$3\times 2x+3\times 3y=3\times 4,2\times 3x+2\times 2y=2\times 7$$

Whakarūnātia.

$$6x+9y=12,6x+4y=14$$

Me tango 6x+4y=14 mai i 6x+9y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+9y-4y=12-14$$

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.

$$9y-4y=12-14$$

Tāpiri 9y ki te -4y.

$$5y=12-14$$

Tāpiri 12 ki te -14.

$$5y=-2$$

Whakawehea ngā taha e rua ki te 5.

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

Whakaurua te -\frac{2}{5} mō y ki 3x+2y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-\frac{2}{5}\right)=7$$

Whakareatia 2 ki te -\frac{2}{5}.

$$3x-\frac{4}{5}=7$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$3x=\frac{39}{5}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{13}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{5},y=-\frac{2}{5}$$