Whakaoti i te {l}{5x+2y=16}{x-3y=10}

$$\left. \begin{array} { l } { 5 x + 2 y = 16 } \\ { x - 3 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-2$$

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.

$$5x+2y=16,x-3y=10$$

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

$$5x+2y=16$$

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

$$5x=-2y+16$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+16\right)$$

Whakareatia \frac{1}{5} ki te -2y+16.

$$x=-\frac{2}{5}y+\frac{16}{5}$$

Whakakapia te \frac{-2y+16}{5} mō te x ki tērā atu whārite, x-3y=10.

$$-\frac{2}{5}y+\frac{16}{5}-3y=10$$

Tāpiri -\frac{2y}{5} ki te -3y.

$$-\frac{17}{5}y+\frac{16}{5}=10$$

Me tango \frac{16}{5} mai i ngā taha e rua o te whārite.

$$-\frac{17}{5}y=\frac{34}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{2}{5}y+\frac{16}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\left(-2\right)+\frac{16}{5}$$

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

$$x=\frac{4+16}{5}$$

Tāpiri \frac{16}{5} ki te \frac{4}{5} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-2$$

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.

$$5x+2y=16,x-3y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\1&-3\end{matrix}\right))\left(\begin{matrix}5&2\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\1&-3\end{matrix}\right))\left(\begin{matrix}16\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-2}&-\frac{2}{5\left(-3\right)-2}\\-\frac{1}{5\left(-3\right)-2}&\frac{5}{5\left(-3\right)-2}\end{matrix}\right)\left(\begin{matrix}16\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}&\frac{2}{17}\\\frac{1}{17}&-\frac{5}{17}\end{matrix}\right)\left(\begin{matrix}16\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}\times 16+\frac{2}{17}\times 10\\\frac{1}{17}\times 16-\frac{5}{17}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-2$$

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.

$$5x+2y=16,x-3y=10$$

Kia ōrite ai a 5x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$5x+2y=16,5x+5\left(-3\right)y=5\times 10$$

Whakarūnātia.

$$5x+2y=16,5x-15y=50$$

Me tango 5x-15y=50 mai i 5x+2y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+2y+15y=16-50$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+15y=16-50$$

Tāpiri 2y ki te 15y.

$$17y=16-50$$

Tāpiri 16 ki te -50.

$$17y=-34$$

Whakawehea ngā taha e rua ki te 17.

$$y=-2$$

Whakaurua te -2 mō y ki x-3y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\left(-2\right)=10$$

Whakareatia -3 ki te -2.

$$x+6=10$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-2$$