Whakaoti i te {l}{-x+y=2}{2y=9x+9}

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

Whakaoti mō x, y

$$x=-\frac{5}{7}\approx -0.714285714$$

$$y = \frac{9}{7} = 1\frac{2}{7} \approx 1.285714286$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$2y-9x=9$$

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.

$$-x+y=2,-9x+2y=9$$

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

$$-x+y=2$$

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

$$-x=-y+2$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-y+2\right)$$

Whakareatia -1 ki te -y+2.

$$x=y-2$$

Whakakapia te y-2 mō te x ki tērā atu whārite, -9x+2y=9.

$$-9\left(y-2\right)+2y=9$$

Whakareatia -9 ki te y-2.

$$-9y+18+2y=9$$

Tāpiri -9y ki te 2y.

$$-7y+18=9$$

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

$$-7y=-9$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{9}{7}$$

Whakaurua te \frac{9}{7} mō y ki x=y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{9}{7}-2$$

Tāpiri -2 ki te \frac{9}{7}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{7},y=\frac{9}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$2y-9x=9$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-x+y=2,-9x+2y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&1\\-9&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&1\\-9&2\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-2-\left(-9\right)}&-\frac{1}{-2-\left(-9\right)}\\-\frac{-9}{-2-\left(-9\right)}&-\frac{1}{-2-\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}2\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&-\frac{1}{7}\\\frac{9}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}2\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 2-\frac{1}{7}\times 9\\\frac{9}{7}\times 2-\frac{1}{7}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{7}\\\frac{9}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{7},y=\frac{9}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$2y-9x=9$$

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.

$$-x+y=2,-9x+2y=9$$

Kia ōrite ai a -x me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$-9\left(-1\right)x-9y=-9\times 2,-\left(-9\right)x-2y=-9$$

Whakarūnātia.

$$9x-9y=-18,9x-2y=-9$$

Me tango 9x-2y=-9 mai i 9x-9y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x-9y+2y=-18+9$$

Tāpiri 9x ki te -9x. Ka whakakore atu ngā kupu 9x me -9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y+2y=-18+9$$

Tāpiri -9y ki te 2y.

$$-7y=-18+9$$

Tāpiri -18 ki te 9.

$$-7y=-9$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{9}{7}$$

Whakaurua te \frac{9}{7} mō y ki -9x+2y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x+2\times \frac{9}{7}=9$$

Whakareatia 2 ki te \frac{9}{7}.

$$-9x+\frac{18}{7}=9$$

Me tango \frac{18}{7} mai i ngā taha e rua o te whārite.

$$-9x=\frac{45}{7}$$

Whakawehea ngā taha e rua ki te -9.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{7},y=\frac{9}{7}$$