Whakaoti i te {l}{-7x-2y=14}{6x+6y=18}

$$\left\{ \begin{array} { l } { - 7 x - 2 y = 14 } \\ { 6 x + 6 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=7$$

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.

$$-7x-2y=14,6x+6y=18$$

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

$$-7x-2y=14$$

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

$$-7x=2y+14$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(2y+14\right)$$

Whakareatia -\frac{1}{7} ki te 14+2y.

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

Whakakapia te -\frac{2y}{7}-2 mō te x ki tērā atu whārite, 6x+6y=18.

$$6\left(-\frac{2}{7}y-2\right)+6y=18$$

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

$$-\frac{12}{7}y-12+6y=18$$

Tāpiri -\frac{12y}{7} ki te 6y.

$$\frac{30}{7}y-12=18$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$\frac{30}{7}y=30$$

Whakawehea ngā taha e rua o te whārite ki te \frac{30}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

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

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

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

$$x=-2-2$$

Tāpiri -2 ki te -2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=7$$

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.

$$-7x-2y=14,6x+6y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right))\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right))\left(\begin{matrix}14\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right))\left(\begin{matrix}14\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\6&6\end{matrix}\right))\left(\begin{matrix}14\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{-7\times 6-\left(-2\times 6\right)}&-\frac{-2}{-7\times 6-\left(-2\times 6\right)}\\-\frac{6}{-7\times 6-\left(-2\times 6\right)}&-\frac{7}{-7\times 6-\left(-2\times 6\right)}\end{matrix}\right)\left(\begin{matrix}14\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&-\frac{1}{15}\\\frac{1}{5}&\frac{7}{30}\end{matrix}\right)\left(\begin{matrix}14\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 14-\frac{1}{15}\times 18\\\frac{1}{5}\times 14+\frac{7}{30}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=7$$

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.

$$-7x-2y=14,6x+6y=18$$

Kia ōrite ai a -7x 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 -7.

$$6\left(-7\right)x+6\left(-2\right)y=6\times 14,-7\times 6x-7\times 6y=-7\times 18$$

Whakarūnātia.

$$-42x-12y=84,-42x-42y=-126$$

Me tango -42x-42y=-126 mai i -42x-12y=84 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-42x+42x-12y+42y=84+126$$

Tāpiri -42x ki te 42x. Ka whakakore atu ngā kupu -42x me 42x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y+42y=84+126$$

Tāpiri -12y ki te 42y.

$$30y=84+126$$

Tāpiri 84 ki te 126.

$$30y=210$$

Whakawehea ngā taha e rua ki te 30.

$$y=7$$

Whakaurua te 7 mō y ki 6x+6y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+6\times 7=18$$

Whakareatia 6 ki te 7.

$$6x+42=18$$

Me tango 42 mai i 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.

$$x=-4,y=7$$