Whakaoti i te {l}{y-x=6}{2x+2y=26}

$$\left\{ \begin{array} { l } { y - x = 6 } \\ { 2 x + 2 y = 26 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$y = \frac{19}{2} = 9\frac{1}{2} = 9.5$$

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.

$$y-x=6,2y+2x=26$$

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

$$y-x=6$$

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

$$y=x+6$$

Whakakapia te x+6 mō te y ki tērā atu whārite, 2y+2x=26.

$$2\left(x+6\right)+2x=26$$

Whakareatia 2 ki te x+6.

$$2x+12+2x=26$$

Tāpiri 2x ki te 2x.

$$4x+12=26$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$4x=14$$

Whakawehea ngā taha e rua ki te 4.

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

Whakaurua te \frac{7}{2} mō x ki y=x+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{7}{2}+6$$

Tāpiri 6 ki te \frac{7}{2}.

$$y=\frac{19}{2}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{19}{2},x=\frac{7}{2}$$

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.

$$y-x=6,2y+2x=26$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\2&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\26\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\2&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}1&-1\\2&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}6\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}6\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&2\end{matrix}\right))\left(\begin{matrix}6\\26\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-2\right)}&-\frac{-1}{2-\left(-2\right)}\\-\frac{2}{2-\left(-2\right)}&\frac{1}{2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}6\\26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{4}\\-\frac{1}{2}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\26\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 6+\frac{1}{4}\times 26\\-\frac{1}{2}\times 6+\frac{1}{4}\times 26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{19}{2}\\\frac{7}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{19}{2},x=\frac{7}{2}$$

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.

$$y-x=6,2y+2x=26$$

Kia ōrite ai a y me 2y, 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 1.

$$2y+2\left(-1\right)x=2\times 6,2y+2x=26$$

Whakarūnātia.

$$2y-2x=12,2y+2x=26$$

Me tango 2y+2x=26 mai i 2y-2x=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-2x-2x=12-26$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2x-2x=12-26$$

Tāpiri -2x ki te -2x.

$$-4x=12-26$$

Tāpiri 12 ki te -26.

$$-4x=-14$$

Whakawehea ngā taha e rua ki te -4.

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

Whakaurua te \frac{7}{2} mō x ki 2y+2x=26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+2\times \frac{7}{2}=26$$

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

$$2y+7=26$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$2y=19$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{19}{2}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{19}{2},x=\frac{7}{2}$$