Whakaoti i te {l}{2x+3y=6}{6x+5y=9}

$$\left. \begin{array} { l } { 2 x + 3 y = 6 } \\ { 6 x + 5 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{3}{8}=-0.375$$

$$y = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

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.

$$2x+3y=6,6x+5y=9$$

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

$$2x+3y=6$$

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

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

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+6\right)$$

Whakareatia \frac{1}{2} ki te -3y+6.

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

Whakakapia te -\frac{3y}{2}+3 mō te x ki tērā atu whārite, 6x+5y=9.

$$6\left(-\frac{3}{2}y+3\right)+5y=9$$

Whakareatia 6 ki te -\frac{3y}{2}+3.

$$-9y+18+5y=9$$

Tāpiri -9y ki te 5y.

$$-4y+18=9$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-4y=-9$$

Whakawehea ngā taha e rua ki te -4.

$$y=\frac{9}{4}$$

Whakaurua te \frac{9}{4} mō y ki x=-\frac{3}{2}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{9}{4}+3$$

Whakareatia -\frac{3}{2} ki te \frac{9}{4} 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.

$$x=-\frac{27}{8}+3$$

Tāpiri 3 ki te -\frac{27}{8}.

$$x=-\frac{3}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{8},y=\frac{9}{4}$$

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.

$$2x+3y=6,6x+5y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\6&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\6&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\6&5\end{matrix}\right))\left(\begin{matrix}2&3\\6&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&5\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&5\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&5\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-3\times 6}&-\frac{3}{2\times 5-3\times 6}\\-\frac{6}{2\times 5-3\times 6}&\frac{2}{2\times 5-3\times 6}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8}&\frac{3}{8}\\\frac{3}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{8}\times 6+\frac{3}{8}\times 9\\\frac{3}{4}\times 6-\frac{1}{4}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{8}\\\frac{9}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{3}{8},y=\frac{9}{4}$$

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.

$$2x+3y=6,6x+5y=9$$

Kia ōrite ai a 2x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$6\times 2x+6\times 3y=6\times 6,2\times 6x+2\times 5y=2\times 9$$

Whakarūnātia.

$$12x+18y=36,12x+10y=18$$

Me tango 12x+10y=18 mai i 12x+18y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+18y-10y=36-18$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-10y=36-18$$

Tāpiri 18y ki te -10y.

$$8y=36-18$$

Tāpiri 36 ki te -18.

$$8y=18$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{9}{4}$$

Whakaurua te \frac{9}{4} mō y ki 6x+5y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+5\times \frac{9}{4}=9$$

Whakareatia 5 ki te \frac{9}{4}.

$$6x+\frac{45}{4}=9$$

Me tango \frac{45}{4} mai i ngā taha e rua o te whārite.

$$6x=-\frac{9}{4}$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{3}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{8},y=\frac{9}{4}$$