Whakaoti i te {l}{9x-7y=-19}{3x+y=7}

$$\left. \begin{array} { l } { 9 x - 7 y = - 19 } \\ { 3 x + y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

$$9x-7y=-19,3x+y=7$$

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

$$9x-7y=-19$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$9x=7y-19$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(7y-19\right)$$

Whakareatia \frac{1}{9} ki te 7y-19.

$$x=\frac{7}{9}y-\frac{19}{9}$$

Whakakapia te \frac{7y-19}{9} mō te x ki tērā atu whārite, 3x+y=7.

$$3\left(\frac{7}{9}y-\frac{19}{9}\right)+y=7$$

Whakareatia 3 ki te \frac{7y-19}{9}.

$$\frac{7}{3}y-\frac{19}{3}+y=7$$

Tāpiri \frac{7y}{3} ki te y.

$$\frac{10}{3}y-\frac{19}{3}=7$$

Me tāpiri \frac{19}{3} ki ngā taha e rua o te whārite.

$$\frac{10}{3}y=\frac{40}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{10}{3}, 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{7}{9}y-\frac{19}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{9}\times 4-\frac{19}{9}$$

Whakareatia \frac{7}{9} ki te 4.

$$x=\frac{28-19}{9}$$

Tāpiri -\frac{19}{9} ki te \frac{28}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

$$9x-7y=-19,3x+y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-7\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-19\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-7\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-7\\3&1\end{matrix}\right))\left(\begin{matrix}9&-7\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-7\\3&1\end{matrix}\right))\left(\begin{matrix}-19\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-7\\3&1\end{matrix}\right))\left(\begin{matrix}-19\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-7\\3&1\end{matrix}\right))\left(\begin{matrix}-19\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9-\left(-7\times 3\right)}&-\frac{-7}{9-\left(-7\times 3\right)}\\-\frac{3}{9-\left(-7\times 3\right)}&\frac{9}{9-\left(-7\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-19\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}&\frac{7}{30}\\-\frac{1}{10}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}-19\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}\left(-19\right)+\frac{7}{30}\times 7\\-\frac{1}{10}\left(-19\right)+\frac{3}{10}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

$$9x-7y=-19,3x+y=7$$

Kia ōrite ai a 9x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$3\times 9x+3\left(-7\right)y=3\left(-19\right),9\times 3x+9y=9\times 7$$

Whakarūnātia.

$$27x-21y=-57,27x+9y=63$$

Me tango 27x+9y=63 mai i 27x-21y=-57 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27x-27x-21y-9y=-57-63$$

Tāpiri 27x ki te -27x. Ka whakakore atu ngā kupu 27x me -27x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-21y-9y=-57-63$$

Tāpiri -21y ki te -9y.

$$-30y=-57-63$$

Tāpiri -57 ki te -63.

$$-30y=-120$$

Whakawehea ngā taha e rua ki te -30.

$$y=4$$

Whakaurua te 4 mō y ki 3x+y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4=7$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$