Whakaoti i te {l}{x+2y=2}{x-3y=-5}

$$\left\{ \begin{array} { l } { x + 2 y = 2 } \\ { x - 3 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{4}{5}=-0.8$$

$$y = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

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.

$$x+2y=2,x-3y=-5$$

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

$$x+2y=2$$

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

$$x=-2y+2$$

Whakakapia te -2y+2 mō te x ki tērā atu whārite, x-3y=-5.

$$-2y+2-3y=-5$$

Tāpiri -2y ki te -3y.

$$-5y+2=-5$$

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

$$-5y=-7$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{7}{5}$$

Whakaurua te \frac{7}{5} mō y ki x=-2y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{7}{5}+2$$

Whakareatia -2 ki te \frac{7}{5}.

$$x=-\frac{14}{5}+2$$

Tāpiri 2 ki te -\frac{14}{5}.

$$x=-\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{5},y=\frac{7}{5}$$

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.

$$x+2y=2,x-3y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-3\end{matrix}\right))\left(\begin{matrix}2\\-5\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5}\\\frac{7}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{4}{5},y=\frac{7}{5}$$

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.

$$x+2y=2,x-3y=-5$$

Me tango x-3y=-5 mai i x+2y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+2y+3y=2+5$$

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.

$$2y+3y=2+5$$

Tāpiri 2y ki te 3y.

$$5y=2+5$$

Tāpiri 2 ki te 5.

$$5y=7$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{7}{5}$$

Whakaurua te \frac{7}{5} mō y ki x-3y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times \frac{7}{5}=-5$$

Whakareatia -3 ki te \frac{7}{5}.

$$x-\frac{21}{5}=-5$$

Me tāpiri \frac{21}{5} ki ngā taha e rua o te whārite.

$$x=-\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{5},y=\frac{7}{5}$$