Whakaoti i te {l}{9x-4y=7}{x-4y=-17}

$$\left. \begin{array} { l } { 9 x - 4 y = 7 } \\ { x - 4 y = - 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=5$$

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.

$$9x-4y=7,x-4y=-17$$

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

$$9x-4y=7$$

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

$$9x=4y+7$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(4y+7\right)$$

Whakareatia \frac{1}{9} ki te 4y+7.

$$x=\frac{4}{9}y+\frac{7}{9}$$

Whakakapia te \frac{4y+7}{9} mō te x ki tērā atu whārite, x-4y=-17.

$$\frac{4}{9}y+\frac{7}{9}-4y=-17$$

Tāpiri \frac{4y}{9} ki te -4y.

$$-\frac{32}{9}y+\frac{7}{9}=-17$$

Me tango \frac{7}{9} mai i ngā taha e rua o te whārite.

$$-\frac{32}{9}y=-\frac{160}{9}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{32}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{4}{9}y+\frac{7}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{9}\times 5+\frac{7}{9}$$

Whakareatia \frac{4}{9} ki te 5.

$$x=\frac{20+7}{9}$$

Tāpiri \frac{7}{9} ki te \frac{20}{9} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=5$$

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.

$$9x-4y=7,x-4y=-17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\-17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-4\\1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right))\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right))\left(\begin{matrix}7\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right))\left(\begin{matrix}7\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\1&-4\end{matrix}\right))\left(\begin{matrix}7\\-17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{9\left(-4\right)-\left(-4\right)}&-\frac{-4}{9\left(-4\right)-\left(-4\right)}\\-\frac{1}{9\left(-4\right)-\left(-4\right)}&\frac{9}{9\left(-4\right)-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}7\\-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&-\frac{1}{8}\\\frac{1}{32}&-\frac{9}{32}\end{matrix}\right)\left(\begin{matrix}7\\-17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 7-\frac{1}{8}\left(-17\right)\\\frac{1}{32}\times 7-\frac{9}{32}\left(-17\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=5$$

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.

$$9x-4y=7,x-4y=-17$$

Me tango x-4y=-17 mai i 9x-4y=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-x-4y+4y=7+17$$

Tāpiri -4y ki te 4y. Ka whakakore atu ngā kupu -4y me 4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9x-x=7+17$$

Tāpiri 9x ki te -x.

$$8x=7+17$$

Tāpiri 7 ki te 17.

$$8x=24$$

Whakawehea ngā taha e rua ki te 8.

$$x=3$$

Whakaurua te 3 mō x ki x-4y=-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3-4y=-17$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-4y=-20$$

Kua oti te pūnaha te whakatau.

$$x=3,y=5$$