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

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

Whakaoti mō x, y

$$x=\frac{3x_{6}-x_{3}}{8}$$

$$y=\frac{3x_{3}-x_{6}}{8}$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+y=x_{6}$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3y+x=x_{3}$$

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.

$$3x+y=x_{6},x+3y=x_{3}$$

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

$$3x+y=x_{6}$$

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

$$3x=-y+x_{6}$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

Tāpiri -\frac{y}{3} ki te 3y.

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

Me tango \frac{x_{6}}{3} mai i ngā taha e rua o te whārite.

$$\frac{8}{3}y=-\frac{x_{6}}{3}+x_{3}$$

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.

$$y=\frac{3x_{3}-x_{6}}{8}$$

Whakaurua te \frac{3x_{3}-x_{6}}{8} mō y ki x=-\frac{1}{3}y+\frac{x_{6}}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{3x_{3}-x_{6}}{8}+\frac{x_{6}}{3}$$

Whakareatia -\frac{1}{3} ki te \frac{3x_{3}-x_{6}}{8}.

$$x=\frac{x_{6}}{24}-\frac{x_{3}}{8}+\frac{x_{6}}{3}$$

Tāpiri \frac{x_{6}}{3} ki te -\frac{x_{3}}{8}+\frac{x_{6}}{24}.

$$x=\frac{3x_{6}-x_{3}}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3x_{6}-x_{3}}{8},y=\frac{3x_{3}-x_{6}}{8}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+y=x_{6}$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3y+x=x_{3}$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+y=x_{6},x+3y=x_{3}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3x_{6}-x_{3}}{8},y=\frac{3x_{3}-x_{6}}{8}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+y=x_{6}$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3y+x=x_{3}$$

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.

$$3x+y=x_{6},x+3y=x_{3}$$

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.

$$3x+y=x_{6},3x+3\times 3y=3x_{3}$$

Whakarūnātia.

$$3x+y=x_{6},3x+9y=3x_{3}$$

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

$$3x-3x+y-9y=x_{6}-3x_{3}$$

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.

$$y-9y=x_{6}-3x_{3}$$

Tāpiri y ki te -9y.

$$-8y=x_{6}-3x_{3}$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{3x_{3}-x_{6}}{8}$$

Whakaurua te \frac{-x_{6}+3x_{3}}{8} mō y ki x+3y=x_{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times \frac{3x_{3}-x_{6}}{8}=x_{3}$$

Whakareatia 3 ki te \frac{-x_{6}+3x_{3}}{8}.

$$x+\frac{9x_{3}-3x_{6}}{8}=x_{3}$$

Me tango \frac{-3x_{6}+9x_{3}}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{3x_{6}-x_{3}}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3x_{6}-x_{3}}{8},y=\frac{3x_{3}-x_{6}}{8}$$