Whakaoti i te {l}{3x+y=6}{x+3y=6}

$$\left\{ \begin{array} { l } { 3 x + y = 6 } \\ { x + 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$3x+y=6,x+3y=6$$

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

$$3x+y=6$$

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

$$3x=-y+6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+6\right)$$

Whakareatia \frac{1}{3} ki te -y+6.

$$x=-\frac{1}{3}y+2$$

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

$$-\frac{1}{3}y+2+3y=6$$

Tāpiri -\frac{y}{3} ki te 3y.

$$\frac{8}{3}y+2=6$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$\frac{8}{3}y=4$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō y ki x=-\frac{1}{3}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{3}{2}+2$$

Whakareatia -\frac{1}{3} ki te \frac{3}{2} 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.

$$x=-\frac{1}{2}+2$$

Tāpiri 2 ki te -\frac{1}{2}.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{3}{2}$$

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.

$$3x+y=6,x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&-\frac{1}{8}\\-\frac{1}{8}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}6\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 6-\frac{1}{8}\times 6\\-\frac{1}{8}\times 6+\frac{3}{8}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{2},y=\frac{3}{2}$$

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.

$$3x+y=6,x+3y=6$$

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.

$$3x+y=6,3x+3\times 3y=3\times 6$$

Whakarūnātia.

$$3x+y=6,3x+9y=18$$

Me tango 3x+9y=18 mai i 3x+y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+y-9y=6-18$$

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.

$$y-9y=6-18$$

Tāpiri y ki te -9y.

$$-8y=6-18$$

Tāpiri 6 ki te -18.

$$-8y=-12$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{3}{2}$$

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

$$x+3\times \frac{3}{2}=6$$

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

$$x+\frac{9}{2}=6$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{3}{2}$$