Whakaoti i te {l}{-10x-6y=12}{4x+7y=-14}

$$\left. \begin{array} { l } { - 10 x - 6 y = 12 } \\ { 4 x + 7 y = - 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

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.

$$-10x-6y=12,4x+7y=-14$$

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

$$-10x-6y=12$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$-10x=6y+12$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(6y+12\right)$$

Whakareatia -\frac{1}{10} ki te 12+6y.

$$x=-\frac{3}{5}y-\frac{6}{5}$$

Whakakapia te \frac{-3y-6}{5} mō te x ki tērā atu whārite, 4x+7y=-14.

$$4\left(-\frac{3}{5}y-\frac{6}{5}\right)+7y=-14$$

Whakareatia 4 ki te \frac{-3y-6}{5}.

$$-\frac{12}{5}y-\frac{24}{5}+7y=-14$$

Tāpiri -\frac{12y}{5} ki te 7y.

$$\frac{23}{5}y-\frac{24}{5}=-14$$

Me tāpiri \frac{24}{5} ki ngā taha e rua o te whārite.

$$\frac{23}{5}y=-\frac{46}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{3}{5}y-\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\left(-2\right)-\frac{6}{5}$$

Whakareatia -\frac{3}{5} ki te -2.

$$x=\frac{6-6}{5}$$

Tāpiri -\frac{6}{5} ki te \frac{6}{5} 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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

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.

$$-10x-6y=12,4x+7y=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\-14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&-6\\4&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right))\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right))\left(\begin{matrix}12\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right))\left(\begin{matrix}12\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-6\\4&7\end{matrix}\right))\left(\begin{matrix}12\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{-10\times 7-\left(-6\times 4\right)}&-\frac{-6}{-10\times 7-\left(-6\times 4\right)}\\-\frac{4}{-10\times 7-\left(-6\times 4\right)}&-\frac{10}{-10\times 7-\left(-6\times 4\right)}\end{matrix}\right)\left(\begin{matrix}12\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{46}&-\frac{3}{23}\\\frac{2}{23}&\frac{5}{23}\end{matrix}\right)\left(\begin{matrix}12\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{46}\times 12-\frac{3}{23}\left(-14\right)\\\frac{2}{23}\times 12+\frac{5}{23}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

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.

$$-10x-6y=12,4x+7y=-14$$

Kia ōrite ai a -10x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -10.

$$4\left(-10\right)x+4\left(-6\right)y=4\times 12,-10\times 4x-10\times 7y=-10\left(-14\right)$$

Whakarūnātia.

$$-40x-24y=48,-40x-70y=140$$

Me tango -40x-70y=140 mai i -40x-24y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-40x+40x-24y+70y=48-140$$

Tāpiri -40x ki te 40x. Ka whakakore atu ngā kupu -40x me 40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y+70y=48-140$$

Tāpiri -24y ki te 70y.

$$46y=48-140$$

Tāpiri 48 ki te -140.

$$46y=-92$$

Whakawehea ngā taha e rua ki te 46.

$$y=-2$$

Whakaurua te -2 mō y ki 4x+7y=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+7\left(-2\right)=-14$$

Whakareatia 7 ki te -2.

$$4x-14=-14$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$4x=0$$

Whakawehea ngā taha e rua ki te 4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$