Whakaoti i te {l}{3x-2y=60}{2x+3y=17.2}

$$\left\{ \begin{array} { l } { 3 x - 2 y = 60 } \\ { 2 x + 3 y = 17.2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{1072}{65} = 16\frac{32}{65} \approx 16.492307692$$

$$y = -\frac{342}{65} = -5\frac{17}{65} \approx -5.261538462$$

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.

$$3x-2y=60,2x+3y=17.2$$

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

$$3x-2y=60$$

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

$$3x=2y+60$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y+60\right)$$

Whakareatia \frac{1}{3} ki te 60+2y.

$$x=\frac{2}{3}y+20$$

Whakakapia te \frac{2y}{3}+20 mō te x ki tērā atu whārite, 2x+3y=17.2.

$$2\left(\frac{2}{3}y+20\right)+3y=17.2$$

Whakareatia 2 ki te \frac{2y}{3}+20.

$$\frac{4}{3}y+40+3y=17.2$$

Tāpiri \frac{4y}{3} ki te 3y.

$$\frac{13}{3}y+40=17.2$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$\frac{13}{3}y=-22.8$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{342}{65}$$

Whakaurua te -\frac{342}{65} mō y ki x=\frac{2}{3}y+20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-\frac{342}{65}\right)+20$$

Whakareatia \frac{2}{3} ki te -\frac{342}{65} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{228}{65}+20$$

Tāpiri 20 ki te -\frac{228}{65}.

$$x=\frac{1072}{65}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1072}{65},y=-\frac{342}{65}$$

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.

$$3x-2y=60,2x+3y=17.2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\2&3\end{matrix}\right))\left(\begin{matrix}3&-2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&3\end{matrix}\right))\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&3\end{matrix}\right))\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&3\end{matrix}\right))\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-2\times 2\right)}&-\frac{-2}{3\times 3-\left(-2\times 2\right)}\\-\frac{2}{3\times 3-\left(-2\times 2\right)}&\frac{3}{3\times 3-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&\frac{2}{13}\\-\frac{2}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}60\\17.2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}\times 60+\frac{2}{13}\times 17.2\\-\frac{2}{13}\times 60+\frac{3}{13}\times 17.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1072}{65}\\-\frac{342}{65}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1072}{65},y=-\frac{342}{65}$$

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.

$$3x-2y=60,2x+3y=17.2$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-2\right)y=2\times 60,3\times 2x+3\times 3y=3\times 17.2$$

Whakarūnātia.

$$6x-4y=120,6x+9y=51.6$$

Me tango 6x+9y=51.6 mai i 6x-4y=120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y-9y=120-51.6$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-9y=120-51.6$$

Tāpiri -4y ki te -9y.

$$-13y=120-51.6$$

Tāpiri 120 ki te -51.6.

$$-13y=68.4$$

Whakawehea ngā taha e rua ki te -13.

$$y=-\frac{342}{65}$$

Whakaurua te -\frac{342}{65} mō y ki 2x+3y=17.2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-\frac{342}{65}\right)=17.2$$

Whakareatia 3 ki te -\frac{342}{65}.

$$2x-\frac{1026}{65}=17.2$$

Me tāpiri \frac{1026}{65} ki ngā taha e rua o te whārite.

$$2x=\frac{2144}{65}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1072}{65}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1072}{65},y=-\frac{342}{65}$$