Whakaoti i te {l}{3x-2y+3=0}{4x+3y-47=0}

$$\left. \begin{array} { l } { 3 x - 2 y + 3 = 0 } \\ { 4 x + 3 y - 47 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=9$$

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.

$$3x-2y+3=0,4x+3y-47=0$$

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

$$3x-2y+3=0$$

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

$$3x-2y=-3$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y-3\right)$$

Whakareatia \frac{1}{3} ki te 2y-3.

$$x=\frac{2}{3}y-1$$

Whakakapia te \frac{2y}{3}-1 mō te x ki tērā atu whārite, 4x+3y-47=0.

$$4\left(\frac{2}{3}y-1\right)+3y-47=0$$

Whakareatia 4 ki te \frac{2y}{3}-1.

$$\frac{8}{3}y-4+3y-47=0$$

Tāpiri \frac{8y}{3} ki te 3y.

$$\frac{17}{3}y-4-47=0$$

Tāpiri -4 ki te -47.

$$\frac{17}{3}y-51=0$$

Me tāpiri 51 ki ngā taha e rua o te whārite.

$$\frac{17}{3}y=51$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=9$$

Whakaurua te 9 mō y ki x=\frac{2}{3}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times 9-1$$

Whakareatia \frac{2}{3} ki te 9.

$$x=6-1$$

Tāpiri -1 ki te 6.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$

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.

$$3x-2y+3=0,4x+3y-47=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\47\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\4&3\end{matrix}\right))\left(\begin{matrix}3&-2\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&3\end{matrix}\right))\left(\begin{matrix}-3\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&3\end{matrix}\right))\left(\begin{matrix}-3\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&3\end{matrix}\right))\left(\begin{matrix}-3\\47\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-2\times 4\right)}&-\frac{-2}{3\times 3-\left(-2\times 4\right)}\\-\frac{4}{3\times 3-\left(-2\times 4\right)}&\frac{3}{3\times 3-\left(-2\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-3\\47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}&\frac{2}{17}\\-\frac{4}{17}&\frac{3}{17}\end{matrix}\right)\left(\begin{matrix}-3\\47\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}\left(-3\right)+\frac{2}{17}\times 47\\-\frac{4}{17}\left(-3\right)+\frac{3}{17}\times 47\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=9$$

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.

$$3x-2y+3=0,4x+3y-47=0$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\left(-2\right)y+4\times 3=0,3\times 4x+3\times 3y+3\left(-47\right)=0$$

Whakarūnātia.

$$12x-8y+12=0,12x+9y-141=0$$

Me tango 12x+9y-141=0 mai i 12x-8y+12=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-8y-9y+12+141=0$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-9y+12+141=0$$

Tāpiri -8y ki te -9y.

$$-17y+12+141=0$$

Tāpiri 12 ki te 141.

$$-17y+153=0$$

Me tango 153 mai i ngā taha e rua o te whārite.

$$-17y=-153$$

Whakawehea ngā taha e rua ki te -17.

$$y=9$$

Whakaurua te 9 mō y ki 4x+3y-47=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\times 9-47=0$$

Whakareatia 3 ki te 9.

$$4x+27-47=0$$

Tāpiri 27 ki te -47.

$$4x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$4x=20$$

Whakawehea ngā taha e rua ki te 4.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$