Whakaoti i te {l}{5x+2y=17}{2x+3y=3}

$$\left. \begin{array} { l } { 5 x + 2 y = 17 } \\ { 2 x + 3 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{45}{11} = 4\frac{1}{11} \approx 4.090909091$$

$$y = -\frac{19}{11} = -1\frac{8}{11} \approx -1.727272727$$

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.

$$5x+2y=17,2x+3y=3$$

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

$$5x+2y=17$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+17$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+17\right)$$

Whakareatia \frac{1}{5} ki te -2y+17.

$$x=-\frac{2}{5}y+\frac{17}{5}$$

Whakakapia te \frac{-2y+17}{5} mō te x ki tērā atu whārite, 2x+3y=3.

$$2\left(-\frac{2}{5}y+\frac{17}{5}\right)+3y=3$$

Whakareatia 2 ki te \frac{-2y+17}{5}.

$$-\frac{4}{5}y+\frac{34}{5}+3y=3$$

Tāpiri -\frac{4y}{5} ki te 3y.

$$\frac{11}{5}y+\frac{34}{5}=3$$

Me tango \frac{34}{5} mai i ngā taha e rua o te whārite.

$$\frac{11}{5}y=-\frac{19}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{19}{11}$$

Whakaurua te -\frac{19}{11} mō y ki x=-\frac{2}{5}y+\frac{17}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\left(-\frac{19}{11}\right)+\frac{17}{5}$$

Whakareatia -\frac{2}{5} ki te -\frac{19}{11} 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.

$$x=\frac{38}{55}+\frac{17}{5}$$

Tāpiri \frac{17}{5} ki te \frac{38}{55} 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.

$$x=\frac{45}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{45}{11},y=-\frac{19}{11}$$

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.

$$5x+2y=17,2x+3y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\2&3\end{matrix}\right))\left(\begin{matrix}5&2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&3\end{matrix}\right))\left(\begin{matrix}17\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&3\end{matrix}\right))\left(\begin{matrix}17\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\2&3\end{matrix}\right))\left(\begin{matrix}17\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-2\times 2}&-\frac{2}{5\times 3-2\times 2}\\-\frac{2}{5\times 3-2\times 2}&\frac{5}{5\times 3-2\times 2}\end{matrix}\right)\left(\begin{matrix}17\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&-\frac{2}{11}\\-\frac{2}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}17\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 17-\frac{2}{11}\times 3\\-\frac{2}{11}\times 17+\frac{5}{11}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{45}{11}\\-\frac{19}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{45}{11},y=-\frac{19}{11}$$

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.

$$5x+2y=17,2x+3y=3$$

Kia ōrite ai a 5x 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 5.

$$2\times 5x+2\times 2y=2\times 17,5\times 2x+5\times 3y=5\times 3$$

Whakarūnātia.

$$10x+4y=34,10x+15y=15$$

Me tango 10x+15y=15 mai i 10x+4y=34 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+4y-15y=34-15$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-15y=34-15$$

Tāpiri 4y ki te -15y.

$$-11y=34-15$$

Tāpiri 34 ki te -15.

$$-11y=19$$

Whakawehea ngā taha e rua ki te -11.

$$y=-\frac{19}{11}$$

Whakaurua te -\frac{19}{11} mō y ki 2x+3y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-\frac{19}{11}\right)=3$$

Whakareatia 3 ki te -\frac{19}{11}.

$$2x-\frac{57}{11}=3$$

Me tāpiri \frac{57}{11} ki ngā taha e rua o te whārite.

$$2x=\frac{90}{11}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{45}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{45}{11},y=-\frac{19}{11}$$