Whakaoti i te {l}{x-3y+9=0}{3x-2y+1=0}

$$\left. \begin{array} { l } { x - 3 y + 9 = 0 } \\ { 3 x - 2 y + 1 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{15}{7} = 2\frac{1}{7} \approx 2.142857143$$

$$y = \frac{26}{7} = 3\frac{5}{7} \approx 3.714285714$$

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.

$$x-3y+9=0,3x-2y+1=0$$

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

$$x-3y+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x-3y=-9$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y-9$$

Whakakapia te -9+3y mō te x ki tērā atu whārite, 3x-2y+1=0.

$$3\left(3y-9\right)-2y+1=0$$

Whakareatia 3 ki te -9+3y.

$$9y-27-2y+1=0$$

Tāpiri 9y ki te -2y.

$$7y-27+1=0$$

Tāpiri -27 ki te 1.

$$7y-26=0$$

Me tāpiri 26 ki ngā taha e rua o te whārite.

$$7y=26$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{26}{7}$$

Whakaurua te \frac{26}{7} 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.

$$x=3\times \frac{26}{7}-9$$

Whakareatia 3 ki te \frac{26}{7}.

$$x=\frac{78}{7}-9$$

Tāpiri -9 ki te \frac{78}{7}.

$$x=\frac{15}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{7},y=\frac{26}{7}$$

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.

$$x-3y+9=0,3x-2y+1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}&\frac{3}{7}\\-\frac{3}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-9\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\left(-9\right)+\frac{3}{7}\left(-1\right)\\-\frac{3}{7}\left(-9\right)+\frac{1}{7}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{7}\\\frac{26}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{7},y=\frac{26}{7}$$

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.

$$x-3y+9=0,3x-2y+1=0$$

Kia ōrite ai a x 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 1.

$$3x+3\left(-3\right)y+3\times 9=0,3x-2y+1=0$$

Whakarūnātia.

$$3x-9y+27=0,3x-2y+1=0$$

Me tango 3x-2y+1=0 mai i 3x-9y+27=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-9y+2y+27-1=0$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y+2y+27-1=0$$

Tāpiri -9y ki te 2y.

$$-7y+27-1=0$$

Tāpiri 27 ki te -1.

$$-7y+26=0$$

Me tango 26 mai i ngā taha e rua o te whārite.

$$-7y=-26$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{26}{7}$$

Whakaurua te \frac{26}{7} mō y ki 3x-2y+1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times \frac{26}{7}+1=0$$

Whakareatia -2 ki te \frac{26}{7}.

$$3x-\frac{52}{7}+1=0$$

Tāpiri -\frac{52}{7} ki te 1.

$$3x-\frac{45}{7}=0$$

Me tāpiri \frac{45}{7} ki ngā taha e rua o te whārite.

$$3x=\frac{45}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{15}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{7},y=\frac{26}{7}$$