Whakaoti i te {l}{x-2y=17}{7x-6y=47}

$$\left\{ \begin{array} { l } { x - 2 y = 17 } \\ { 7 x - 6 y = 47 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-9$$

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.

$$x-2y=17,7x-6y=47$$

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

$$x-2y=17$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$x=2y+17$$

Whakakapia te 2y+17 mō te x ki tērā atu whārite, 7x-6y=47.

$$7\left(2y+17\right)-6y=47$$

Whakareatia 7 ki te 2y+17.

$$14y+119-6y=47$$

Tāpiri 14y ki te -6y.

$$8y+119=47$$

Me tango 119 mai i ngā taha e rua o te whārite.

$$8y=-72$$

Whakawehea ngā taha e rua ki te 8.

$$y=-9$$

Whakaurua te -9 mō y ki x=2y+17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\left(-9\right)+17$$

Whakareatia 2 ki te -9.

$$x=-18+17$$

Tāpiri 17 ki te -18.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-9$$

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.

$$x-2y=17,7x-6y=47$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\47\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\7&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right))\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right))\left(\begin{matrix}17\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right))\left(\begin{matrix}17\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\7&-6\end{matrix}\right))\left(\begin{matrix}17\\47\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-\left(-2\times 7\right)}&-\frac{-2}{-6-\left(-2\times 7\right)}\\-\frac{7}{-6-\left(-2\times 7\right)}&\frac{1}{-6-\left(-2\times 7\right)}\end{matrix}\right)\left(\begin{matrix}17\\47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4}&\frac{1}{4}\\-\frac{7}{8}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}17\\47\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4}\times 17+\frac{1}{4}\times 47\\-\frac{7}{8}\times 17+\frac{1}{8}\times 47\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-9$$

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.

$$x-2y=17,7x-6y=47$$

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.

$$7x+7\left(-2\right)y=7\times 17,7x-6y=47$$

Whakarūnātia.

$$7x-14y=119,7x-6y=47$$

Me tango 7x-6y=47 mai i 7x-14y=119 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-14y+6y=119-47$$

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.

$$-14y+6y=119-47$$

Tāpiri -14y ki te 6y.

$$-8y=119-47$$

Tāpiri 119 ki te -47.

$$-8y=72$$

Whakawehea ngā taha e rua ki te -8.

$$y=-9$$

Whakaurua te -9 mō y ki 7x-6y=47. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-6\left(-9\right)=47$$

Whakareatia -6 ki te -9.

$$7x+54=47$$

Me tango 54 mai i ngā taha e rua o te whārite.

$$7x=-7$$

Whakawehea ngā taha e rua ki te 7.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-9$$