Whakaoti i te {r}{-2x+9y=8}{x-2y=6}

$$\left. \begin{array} { r } { - 2 x + 9 y = 8 } \\ { x - 2 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=14$$

$$y=4$$

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.

$$-2x+9y=8,x-2y=6$$

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

$$-2x+9y=8$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$-2x=-9y+8$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-9y+8\right)$$

Whakareatia -\frac{1}{2} ki te -9y+8.

$$x=\frac{9}{2}y-4$$

Whakakapia te \frac{9y}{2}-4 mō te x ki tērā atu whārite, x-2y=6.

$$\frac{9}{2}y-4-2y=6$$

Tāpiri \frac{9y}{2} ki te -2y.

$$\frac{5}{2}y-4=6$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=10$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{9}{2}y-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{9}{2}\times 4-4$$

Whakareatia \frac{9}{2} ki te 4.

$$x=18-4$$

Tāpiri -4 ki te 18.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=4$$

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.

$$-2x+9y=8,x-2y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&9\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right))\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&9\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2\left(-2\right)-9}&-\frac{9}{-2\left(-2\right)-9}\\-\frac{1}{-2\left(-2\right)-9}&-\frac{2}{-2\left(-2\right)-9}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&\frac{9}{5}\\\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 8+\frac{9}{5}\times 6\\\frac{1}{5}\times 8+\frac{2}{5}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=14,y=4$$

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.

$$-2x+9y=8,x-2y=6$$

Kia ōrite ai a -2x 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 -2.

$$-2x+9y=8,-2x-2\left(-2\right)y=-2\times 6$$

Whakarūnātia.

$$-2x+9y=8,-2x+4y=-12$$

Me tango -2x+4y=-12 mai i -2x+9y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+9y-4y=8+12$$

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

$$9y-4y=8+12$$

Tāpiri 9y ki te -4y.

$$5y=8+12$$

Tāpiri 8 ki te 12.

$$5y=20$$

Whakawehea ngā taha e rua ki te 5.

$$y=4$$

Whakaurua te 4 mō y ki x-2y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times 4=6$$

Whakareatia -2 ki te 4.

$$x-8=6$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=4$$