Whakaoti i te {l}{6x-4y=30}{2x+6y=-34}

$$\left\{ \begin{array} { l } { 6 x - 4 y = 30 } \\ { 2 x + 6 y = - 34 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-6$$

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.

$$6x-4y=30,2x+6y=-34$$

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

$$6x-4y=30$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$6x=4y+30$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(4y+30\right)$$

Whakareatia \frac{1}{6} ki te 4y+30.

$$x=\frac{2}{3}y+5$$

Whakakapia te \frac{2y}{3}+5 mō te x ki tērā atu whārite, 2x+6y=-34.

$$2\left(\frac{2}{3}y+5\right)+6y=-34$$

Whakareatia 2 ki te \frac{2y}{3}+5.

$$\frac{4}{3}y+10+6y=-34$$

Tāpiri \frac{4y}{3} ki te 6y.

$$\frac{22}{3}y+10=-34$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$\frac{22}{3}y=-44$$

Whakawehea ngā taha e rua o te whārite ki te \frac{22}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

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

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

Whakareatia \frac{2}{3} ki te -6.

$$x=-4+5$$

Tāpiri 5 ki te -4.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$

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.

$$6x-4y=30,2x+6y=-34$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-4\\2&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\-34\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-4\\2&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-4\\2&6\end{matrix}\right))\left(\begin{matrix}6&-4\\2&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-4\\2&6\end{matrix}\right))\left(\begin{matrix}30\\-34\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-4\\2&6\end{matrix}\right))\left(\begin{matrix}30\\-34\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-4\\2&6\end{matrix}\right))\left(\begin{matrix}30\\-34\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{6\times 6-\left(-4\times 2\right)}&-\frac{-4}{6\times 6-\left(-4\times 2\right)}\\-\frac{2}{6\times 6-\left(-4\times 2\right)}&\frac{6}{6\times 6-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}30\\-34\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}&\frac{1}{11}\\-\frac{1}{22}&\frac{3}{22}\end{matrix}\right)\left(\begin{matrix}30\\-34\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}\times 30+\frac{1}{11}\left(-34\right)\\-\frac{1}{22}\times 30+\frac{3}{22}\left(-34\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-6$$

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.

$$6x-4y=30,2x+6y=-34$$

Kia ōrite ai a 6x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$2\times 6x+2\left(-4\right)y=2\times 30,6\times 2x+6\times 6y=6\left(-34\right)$$

Whakarūnātia.

$$12x-8y=60,12x+36y=-204$$

Me tango 12x+36y=-204 mai i 12x-8y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-8y-36y=60+204$$

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.

$$-8y-36y=60+204$$

Tāpiri -8y ki te -36y.

$$-44y=60+204$$

Tāpiri 60 ki te 204.

$$-44y=264$$

Whakawehea ngā taha e rua ki te -44.

$$y=-6$$

Whakaurua te -6 mō y ki 2x+6y=-34. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+6\left(-6\right)=-34$$

Whakareatia 6 ki te -6.

$$2x-36=-34$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$