Whakaoti i te {r}{x+4y=40}{-x+8y=68}

$$\left. \begin{array} { r } { x + 4 y = 40 } \\ { - x + 8 y = 68 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=9$$

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.

$$x+4y=40,-x+8y=68$$

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

$$x+4y=40$$

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

$$x=-4y+40$$

Whakakapia te -4y+40 mō te x ki tērā atu whārite, -x+8y=68.

$$-\left(-4y+40\right)+8y=68$$

Whakareatia -1 ki te -4y+40.

$$4y-40+8y=68$$

Tāpiri 4y ki te 8y.

$$12y-40=68$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$12y=108$$

Whakawehea ngā taha e rua ki te 12.

$$y=9$$

Whakaurua te 9 mō y ki x=-4y+40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times 9+40$$

Whakareatia -4 ki te 9.

$$x=-36+40$$

Tāpiri 40 ki te -36.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=9$$

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.

$$x+4y=40,-x+8y=68$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\-1&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\68\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\-1&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&4\\-1&8\end{matrix}\right))\left(\begin{matrix}1&4\\-1&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-1&8\end{matrix}\right))\left(\begin{matrix}40\\68\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-1&8\end{matrix}\right))\left(\begin{matrix}40\\68\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-1&8\end{matrix}\right))\left(\begin{matrix}40\\68\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{8-4\left(-1\right)}&-\frac{4}{8-4\left(-1\right)}\\-\frac{-1}{8-4\left(-1\right)}&\frac{1}{8-4\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}40\\68\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&-\frac{1}{3}\\\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}40\\68\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 40-\frac{1}{3}\times 68\\\frac{1}{12}\times 40+\frac{1}{12}\times 68\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=9$$

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.

$$x+4y=40,-x+8y=68$$

Kia ōrite ai a x 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 1.

$$-x-4y=-40,-x+8y=68$$

Me tango -x+8y=68 mai i -x-4y=-40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-4y-8y=-40-68$$

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

$$-4y-8y=-40-68$$

Tāpiri -4y ki te -8y.

$$-12y=-40-68$$

Tāpiri -40 ki te -68.

$$-12y=-108$$

Whakawehea ngā taha e rua ki te -12.

$$y=9$$

Whakaurua te 9 mō y ki -x+8y=68. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+8\times 9=68$$

Whakareatia 8 ki te 9.

$$-x+72=68$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-x=-4$$

Whakawehea ngā taha e rua ki te -1.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=9$$