Whakaoti i te {l}{2x+3y=7}{6x+y=10}

$$\left. \begin{array} { l } { 2 x + 3 y = 7 } \\ { 6 x + y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{23}{16} = 1\frac{7}{16} = 1.4375$$

$$y = \frac{11}{8} = 1\frac{3}{8} = 1.375$$

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.

$$2x+3y=7,6x+y=10$$

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

$$2x+3y=7$$

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

$$2x=-3y+7$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+7\right)$$

Whakareatia \frac{1}{2} ki te -3y+7.

$$x=-\frac{3}{2}y+\frac{7}{2}$$

Whakakapia te \frac{-3y+7}{2} mō te x ki tērā atu whārite, 6x+y=10.

$$6\left(-\frac{3}{2}y+\frac{7}{2}\right)+y=10$$

Whakareatia 6 ki te \frac{-3y+7}{2}.

$$-9y+21+y=10$$

Tāpiri -9y ki te y.

$$-8y+21=10$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$-8y=-11$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{11}{8}$$

Whakaurua te \frac{11}{8} mō y ki x=-\frac{3}{2}y+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{11}{8}+\frac{7}{2}$$

Whakareatia -\frac{3}{2} ki te \frac{11}{8} 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.

$$x=-\frac{33}{16}+\frac{7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{33}{16} 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.

$$x=\frac{23}{16}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{16},y=\frac{11}{8}$$

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.

$$2x+3y=7,6x+y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\6&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\6&1\end{matrix}\right))\left(\begin{matrix}2&3\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&1\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&1\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&1\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-3\times 6}&-\frac{3}{2-3\times 6}\\-\frac{6}{2-3\times 6}&\frac{2}{2-3\times 6}\end{matrix}\right)\left(\begin{matrix}7\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{16}&\frac{3}{16}\\\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}7\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{16}\times 7+\frac{3}{16}\times 10\\\frac{3}{8}\times 7-\frac{1}{8}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{16}\\\frac{11}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{23}{16},y=\frac{11}{8}$$

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.

$$2x+3y=7,6x+y=10$$

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.

$$6\times 2x+6\times 3y=6\times 7,2\times 6x+2y=2\times 10$$

Whakarūnātia.

$$12x+18y=42,12x+2y=20$$

Me tango 12x+2y=20 mai i 12x+18y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+18y-2y=42-20$$

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.

$$18y-2y=42-20$$

Tāpiri 18y ki te -2y.

$$16y=42-20$$

Tāpiri 42 ki te -20.

$$16y=22$$

Whakawehea ngā taha e rua ki te 16.

$$y=\frac{11}{8}$$

Whakaurua te \frac{11}{8} mō y ki 6x+y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+\frac{11}{8}=10$$

Me tango \frac{11}{8} mai i ngā taha e rua o te whārite.

$$6x=\frac{69}{8}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{23}{16}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{23}{16},y=\frac{11}{8}$$