Whakaoti i te {c}{x-y=4}{6x+7y=-80}

$$\left. \begin{array} { c } { x - y = 4 } \\ { 6 x + 7 y = - 80 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-8$$

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.

$$x-y=4,6x+7y=-80$$

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

$$x-y=4$$

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

$$x=y+4$$

Whakakapia te y+4 mō te x ki tērā atu whārite, 6x+7y=-80.

$$6\left(y+4\right)+7y=-80$$

Whakareatia 6 ki te y+4.

$$6y+24+7y=-80$$

Tāpiri 6y ki te 7y.

$$13y+24=-80$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$13y=-104$$

Whakawehea ngā taha e rua ki te 13.

$$y=-8$$

Whakaurua te -8 mō y ki x=y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-8+4$$

Tāpiri 4 ki te -8.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-8$$

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.

$$x-y=4,6x+7y=-80$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\6&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-80\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\6&7\end{matrix}\right))\left(\begin{matrix}1&-1\\6&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&7\end{matrix}\right))\left(\begin{matrix}4\\-80\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&7\end{matrix}\right))\left(\begin{matrix}4\\-80\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&7\end{matrix}\right))\left(\begin{matrix}4\\-80\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-6\right)}&-\frac{-1}{7-\left(-6\right)}\\-\frac{6}{7-\left(-6\right)}&\frac{1}{7-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}4\\-80\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}&\frac{1}{13}\\-\frac{6}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}4\\-80\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}\times 4+\frac{1}{13}\left(-80\right)\\-\frac{6}{13}\times 4+\frac{1}{13}\left(-80\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-8$$

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.

$$x-y=4,6x+7y=-80$$

Kia ōrite ai a x 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 1.

$$6x+6\left(-1\right)y=6\times 4,6x+7y=-80$$

Whakarūnātia.

$$6x-6y=24,6x+7y=-80$$

Me tango 6x+7y=-80 mai i 6x-6y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-6y-7y=24+80$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-7y=24+80$$

Tāpiri -6y ki te -7y.

$$-13y=24+80$$

Tāpiri 24 ki te 80.

$$-13y=104$$

Whakawehea ngā taha e rua ki te -13.

$$y=-8$$

Whakaurua te -8 mō y ki 6x+7y=-80. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+7\left(-8\right)=-80$$

Whakareatia 7 ki te -8.

$$6x-56=-80$$

Me tāpiri 56 ki ngā taha e rua o te whārite.

$$6x=-24$$

Whakawehea ngā taha e rua ki te 6.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-8$$