Whakaoti i te {l}{x+2y=8}{x-3y=9}

$$\left. \begin{array} { l } { x + 2 y = 8 } \\ { x - 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{42}{5} = 8\frac{2}{5} = 8.4$$

$$y=-\frac{1}{5}=-0.2$$

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.

$$x+2y=8,x-3y=9$$

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

$$x+2y=8$$

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

$$x=-2y+8$$

Whakakapia te -2y+8 mō te x ki tērā atu whārite, x-3y=9.

$$-2y+8-3y=9$$

Tāpiri -2y ki te -3y.

$$-5y+8=9$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-5y=1$$

Whakawehea ngā taha e rua ki te -5.

$$y=-\frac{1}{5}$$

Whakaurua te -\frac{1}{5} mō y ki x=-2y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-\frac{1}{5}\right)+8$$

Whakareatia -2 ki te -\frac{1}{5}.

$$x=\frac{2}{5}+8$$

Tāpiri 8 ki te \frac{2}{5}.

$$x=\frac{42}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{42}{5},y=-\frac{1}{5}$$

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.

$$x+2y=8,x-3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\1&-3\end{matrix}\right))\left(\begin{matrix}1&2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-2}&-\frac{2}{-3-2}\\-\frac{1}{-3-2}&\frac{1}{-3-2}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&\frac{2}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 8+\frac{2}{5}\times 9\\\frac{1}{5}\times 8-\frac{1}{5}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{42}{5}\\-\frac{1}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{42}{5},y=-\frac{1}{5}$$

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.

$$x+2y=8,x-3y=9$$

Me tango x-3y=9 mai i x+2y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+2y+3y=8-9$$

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.

$$2y+3y=8-9$$

Tāpiri 2y ki te 3y.

$$5y=8-9$$

Tāpiri 8 ki te -9.

$$5y=-1$$

Whakawehea ngā taha e rua ki te 5.

$$y=-\frac{1}{5}$$

Whakaurua te -\frac{1}{5} mō y ki x-3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\left(-\frac{1}{5}\right)=9$$

Whakareatia -3 ki te -\frac{1}{5}.

$$x+\frac{3}{5}=9$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{42}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{42}{5},y=-\frac{1}{5}$$