Whakaoti i te {l}{2x-3y=-5}{4x+9y=-7}

$$\left\{ \begin{array} { l } { 2 x - 3 y = - 5 } \\ { 4 x + 9 y = - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{11}{5} = -2\frac{1}{5} = -2.2$$

$$y=\frac{1}{5}=0.2$$

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.

$$2x-3y=-5,4x+9y=-7$$

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

$$2x-3y=-5$$

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

$$2x=3y-5$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y-5\right)$$

Whakareatia \frac{1}{2} ki te 3y-5.

$$x=\frac{3}{2}y-\frac{5}{2}$$

Whakakapia te \frac{3y-5}{2} mō te x ki tērā atu whārite, 4x+9y=-7.

$$4\left(\frac{3}{2}y-\frac{5}{2}\right)+9y=-7$$

Whakareatia 4 ki te \frac{3y-5}{2}.

$$6y-10+9y=-7$$

Tāpiri 6y ki te 9y.

$$15y-10=-7$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$15y=3$$

Whakawehea ngā taha e rua ki te 15.

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

Whakaurua te \frac{1}{5} mō y ki x=\frac{3}{2}y-\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times \frac{1}{5}-\frac{5}{2}$$

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

$$x=\frac{3}{10}-\frac{5}{2}$$

Tāpiri -\frac{5}{2} ki te \frac{3}{10} 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.

$$x=-\frac{11}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{5},y=\frac{1}{5}$$

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.

$$2x-3y=-5,4x+9y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\4&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}2&-3\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\times 9-\left(-3\times 4\right)}&-\frac{-3}{2\times 9-\left(-3\times 4\right)}\\-\frac{4}{2\times 9-\left(-3\times 4\right)}&\frac{2}{2\times 9-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{10}\\-\frac{2}{15}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}-5\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\left(-5\right)+\frac{1}{10}\left(-7\right)\\-\frac{2}{15}\left(-5\right)+\frac{1}{15}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{5}\\\frac{1}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{11}{5},y=\frac{1}{5}$$

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.

$$2x-3y=-5,4x+9y=-7$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\left(-3\right)y=4\left(-5\right),2\times 4x+2\times 9y=2\left(-7\right)$$

Whakarūnātia.

$$8x-12y=-20,8x+18y=-14$$

Me tango 8x+18y=-14 mai i 8x-12y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-12y-18y=-20+14$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-18y=-20+14$$

Tāpiri -12y ki te -18y.

$$-30y=-20+14$$

Tāpiri -20 ki te 14.

$$-30y=-6$$

Whakawehea ngā taha e rua ki te -30.

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

Whakaurua te \frac{1}{5} mō y ki 4x+9y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+9\times \frac{1}{5}=-7$$

Whakareatia 9 ki te \frac{1}{5}.

$$4x+\frac{9}{5}=-7$$

Me tango \frac{9}{5} mai i ngā taha e rua o te whārite.

$$4x=-\frac{44}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{11}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{5},y=\frac{1}{5}$$