Whakaoti i te {c}{2x+9y=-7}{6x-3y=9}

$$\left. \begin{array} { c } { 2 x + 9 y = - 7 } \\ { 6 x - 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$2x+9y=-7,6x-3y=9$$

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

$$2x+9y=-7$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$2x=-9y-7$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -9y-7.

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

Whakakapia te \frac{-9y-7}{2} mō te x ki tērā atu whārite, 6x-3y=9.

$$6\left(-\frac{9}{2}y-\frac{7}{2}\right)-3y=9$$

Whakareatia 6 ki te \frac{-9y-7}{2}.

$$-27y-21-3y=9$$

Tāpiri -27y ki te -3y.

$$-30y-21=9$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$-30y=30$$

Whakawehea ngā taha e rua ki te -30.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{9}{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.

$$x=-\frac{9}{2}\left(-1\right)-\frac{7}{2}$$

Whakareatia -\frac{9}{2} ki te -1.

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

Tāpiri -\frac{7}{2} ki te \frac{9}{2} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$

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.

$$2x+9y=-7,6x-3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&9\\6&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&9\\6&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&9\\6&-3\end{matrix}\right))\left(\begin{matrix}2&9\\6&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\6&-3\end{matrix}\right))\left(\begin{matrix}-7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\6&-3\end{matrix}\right))\left(\begin{matrix}-7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\6&-3\end{matrix}\right))\left(\begin{matrix}-7\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-9\times 6}&-\frac{9}{2\left(-3\right)-9\times 6}\\-\frac{6}{2\left(-3\right)-9\times 6}&\frac{2}{2\left(-3\right)-9\times 6}\end{matrix}\right)\left(\begin{matrix}-7\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}&\frac{3}{20}\\\frac{1}{10}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}-7\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}\left(-7\right)+\frac{3}{20}\times 9\\\frac{1}{10}\left(-7\right)-\frac{1}{30}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

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.

$$2x+9y=-7,6x-3y=9$$

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.

$$6\times 2x+6\times 9y=6\left(-7\right),2\times 6x+2\left(-3\right)y=2\times 9$$

Whakarūnātia.

$$12x+54y=-42,12x-6y=18$$

Me tango 12x-6y=18 mai i 12x+54y=-42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+54y+6y=-42-18$$

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.

$$54y+6y=-42-18$$

Tāpiri 54y ki te 6y.

$$60y=-42-18$$

Tāpiri -42 ki te -18.

$$60y=-60$$

Whakawehea ngā taha e rua ki te 60.

$$y=-1$$

Whakaurua te -1 mō y ki 6x-3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-3\left(-1\right)=9$$

Whakareatia -3 ki te -1.

$$6x+3=9$$

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

$$6x=6$$

Whakawehea ngā taha e rua ki te 6.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$