Whakaoti i te {l}{5x+2y=6}{9x+2y=22}

$$\left. \begin{array} { l } { 5 x + 2 y = 6 } \\ { 9 x + 2 y = 22 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-7$$

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.

$$5x+2y=6,9x+2y=22$$

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

$$5x+2y=6$$

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

$$5x=-2y+6$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+6\right)$$

Whakareatia \frac{1}{5} ki te -2y+6.

$$x=-\frac{2}{5}y+\frac{6}{5}$$

Whakakapia te \frac{-2y+6}{5} mō te x ki tērā atu whārite, 9x+2y=22.

$$9\left(-\frac{2}{5}y+\frac{6}{5}\right)+2y=22$$

Whakareatia 9 ki te \frac{-2y+6}{5}.

$$-\frac{18}{5}y+\frac{54}{5}+2y=22$$

Tāpiri -\frac{18y}{5} ki te 2y.

$$-\frac{8}{5}y+\frac{54}{5}=22$$

Me tango \frac{54}{5} mai i ngā taha e rua o te whārite.

$$-\frac{8}{5}y=\frac{56}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-7$$

Whakaurua te -7 mō y ki x=-\frac{2}{5}y+\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\left(-7\right)+\frac{6}{5}$$

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

$$x=\frac{14+6}{5}$$

Tāpiri \frac{6}{5} ki te \frac{14}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-7$$

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.

$$5x+2y=6,9x+2y=22$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\9&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\22\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\9&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\9&2\end{matrix}\right))\left(\begin{matrix}5&2\\9&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\9&2\end{matrix}\right))\left(\begin{matrix}6\\22\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\9&2\end{matrix}\right))\left(\begin{matrix}6\\22\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\9&2\end{matrix}\right))\left(\begin{matrix}6\\22\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-2\times 9}&-\frac{2}{5\times 2-2\times 9}\\-\frac{9}{5\times 2-2\times 9}&\frac{5}{5\times 2-2\times 9}\end{matrix}\right)\left(\begin{matrix}6\\22\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{9}{8}&-\frac{5}{8}\end{matrix}\right)\left(\begin{matrix}6\\22\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 6+\frac{1}{4}\times 22\\\frac{9}{8}\times 6-\frac{5}{8}\times 22\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-7$$

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.

$$5x+2y=6,9x+2y=22$$

Me tango 9x+2y=22 mai i 5x+2y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-9x+2y-2y=6-22$$

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.

$$5x-9x=6-22$$

Tāpiri 5x ki te -9x.

$$-4x=6-22$$

Tāpiri 6 ki te -22.

$$-4x=-16$$

Whakawehea ngā taha e rua ki te -4.

$$x=4$$

Whakaurua te 4 mō x ki 9x+2y=22. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$9\times 4+2y=22$$

Whakareatia 9 ki te 4.

$$36+2y=22$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$2y=-14$$

Whakawehea ngā taha e rua ki te 2.

$$y=-7$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-7$$