Whakaoti i te {c}{3x+4y=12}{x+6y=-16}

$$\left. \begin{array} { c } { 3 x + 4 y = 12 } \\ { x + 6 y = - 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{68}{7} = 9\frac{5}{7} \approx 9.714285714$$

$$y = -\frac{30}{7} = -4\frac{2}{7} \approx -4.285714286$$

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.

$$3x+4y=12,x+6y=-16$$

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

$$3x+4y=12$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$3x=-4y+12$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+12\right)$$

Whakareatia \frac{1}{3} ki te -4y+12.

$$x=-\frac{4}{3}y+4$$

Whakakapia te -\frac{4y}{3}+4 mō te x ki tērā atu whārite, x+6y=-16.

$$-\frac{4}{3}y+4+6y=-16$$

Tāpiri -\frac{4y}{3} ki te 6y.

$$\frac{14}{3}y+4=-16$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\frac{14}{3}y=-20$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te -\frac{30}{7} mō y ki x=-\frac{4}{3}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-\frac{30}{7}\right)+4$$

Whakareatia -\frac{4}{3} ki te -\frac{30}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{40}{7}+4$$

Tāpiri 4 ki te \frac{40}{7}.

$$x=\frac{68}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{68}{7},y=-\frac{30}{7}$$

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.

$$3x+4y=12,x+6y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\1&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\1&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\1&6\end{matrix}\right))\left(\begin{matrix}3&4\\1&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&6\end{matrix}\right))\left(\begin{matrix}12\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&6\end{matrix}\right))\left(\begin{matrix}12\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&6\end{matrix}\right))\left(\begin{matrix}12\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{3\times 6-4}&-\frac{4}{3\times 6-4}\\-\frac{1}{3\times 6-4}&\frac{3}{3\times 6-4}\end{matrix}\right)\left(\begin{matrix}12\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&-\frac{2}{7}\\-\frac{1}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}12\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\times 12-\frac{2}{7}\left(-16\right)\\-\frac{1}{14}\times 12+\frac{3}{14}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{68}{7}\\-\frac{30}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{68}{7},y=-\frac{30}{7}$$

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.

$$3x+4y=12,x+6y=-16$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+4y=12,3x+3\times 6y=3\left(-16\right)$$

Whakarūnātia.

$$3x+4y=12,3x+18y=-48$$

Me tango 3x+18y=-48 mai i 3x+4y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+4y-18y=12+48$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-18y=12+48$$

Tāpiri 4y ki te -18y.

$$-14y=12+48$$

Tāpiri 12 ki te 48.

$$-14y=60$$

Whakawehea ngā taha e rua ki te -14.

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

Whakaurua te -\frac{30}{7} mō y ki x+6y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+6\left(-\frac{30}{7}\right)=-16$$

Whakareatia 6 ki te -\frac{30}{7}.

$$x-\frac{180}{7}=-16$$

Me tāpiri \frac{180}{7} ki ngā taha e rua o te whārite.

$$x=\frac{68}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{68}{7},y=-\frac{30}{7}$$