Whakaoti i te {r}{-x+3y=4}{-7x+12y=12}

$$\left. \begin{array} { r } { - x + 3 y = 4 } \\ { - 7 x + 12 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y = \frac{16}{9} = 1\frac{7}{9} \approx 1.777777778$$

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.

$$-x+3y=4,-7x+12y=12$$

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

$$-x+3y=4$$

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

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

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-3y+4\right)$$

Whakareatia -1 ki te -3y+4.

$$x=3y-4$$

Whakakapia te 3y-4 mō te x ki tērā atu whārite, -7x+12y=12.

$$-7\left(3y-4\right)+12y=12$$

Whakareatia -7 ki te 3y-4.

$$-21y+28+12y=12$$

Tāpiri -21y ki te 12y.

$$-9y+28=12$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-9y=-16$$

Whakawehea ngā taha e rua ki te -9.

$$y=\frac{16}{9}$$

Whakaurua te \frac{16}{9} mō y ki x=3y-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times \frac{16}{9}-4$$

Whakareatia 3 ki te \frac{16}{9}.

$$x=\frac{16}{3}-4$$

Tāpiri -4 ki te \frac{16}{3}.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{3},y=\frac{16}{9}$$

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.

$$-x+3y=4,-7x+12y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&3\\-7&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right))\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-7&12\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{-12-3\left(-7\right)}&-\frac{3}{-12-3\left(-7\right)}\\-\frac{-7}{-12-3\left(-7\right)}&-\frac{1}{-12-3\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}4\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}&-\frac{1}{3}\\\frac{7}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}4\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\times 4-\frac{1}{3}\times 12\\\frac{7}{9}\times 4-\frac{1}{9}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\\\frac{16}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{3},y=\frac{16}{9}$$

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.

$$-x+3y=4,-7x+12y=12$$

Kia ōrite ai a -x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$-7\left(-1\right)x-7\times 3y=-7\times 4,-\left(-7\right)x-12y=-12$$

Whakarūnātia.

$$7x-21y=-28,7x-12y=-12$$

Me tango 7x-12y=-12 mai i 7x-21y=-28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-21y+12y=-28+12$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-21y+12y=-28+12$$

Tāpiri -21y ki te 12y.

$$-9y=-28+12$$

Tāpiri -28 ki te 12.

$$-9y=-16$$

Whakawehea ngā taha e rua ki te -9.

$$y=\frac{16}{9}$$

Whakaurua te \frac{16}{9} mō y ki -7x+12y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+12\times \frac{16}{9}=12$$

Whakareatia 12 ki te \frac{16}{9}.

$$-7x+\frac{64}{3}=12$$

Me tango \frac{64}{3} mai i ngā taha e rua o te whārite.

$$-7x=-\frac{28}{3}$$

Whakawehea ngā taha e rua ki te -7.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{3},y=\frac{16}{9}$$