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

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

Whakaoti mō x, y

$$x=-2$$

$$y=-3$$

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.

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

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, 2x-3y=5.

$$2\left(-\frac{2}{5}y-\frac{16}{5}\right)-3y=5$$

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

$$-\frac{4}{5}y-\frac{32}{5}-3y=5$$

Tāpiri -\frac{4y}{5} ki te -3y.

$$-\frac{19}{5}y-\frac{32}{5}=5$$

Me tāpiri \frac{32}{5} ki ngā taha e rua o te whārite.

$$-\frac{19}{5}y=\frac{57}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 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.

$$x=-\frac{2}{5}\left(-3\right)-\frac{16}{5}$$

Whakareatia -\frac{2}{5} ki te -3.

$$x=\frac{6-16}{5}$$

Tāpiri -\frac{16}{5} ki te \frac{6}{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.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}&\frac{2}{19}\\\frac{2}{19}&-\frac{5}{19}\end{matrix}\right)\left(\begin{matrix}-16\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}\left(-16\right)+\frac{2}{19}\times 5\\\frac{2}{19}\left(-16\right)-\frac{5}{19}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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.

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

Whakarūnātia.

$$10x+4y=-32,10x-15y=25$$

Me tango 10x-15y=25 mai i 10x+4y=-32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+4y+15y=-32-25$$

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.

$$4y+15y=-32-25$$

Tāpiri 4y ki te 15y.

$$19y=-32-25$$

Tāpiri -32 ki te -25.

$$19y=-57$$

Whakawehea ngā taha e rua ki te 19.

$$y=-3$$

Whakaurua te -3 mō y ki 2x-3y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-3\right)=5$$

Whakareatia -3 ki te -3.

$$2x+9=5$$

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

$$2x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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