Whakaoti i te {l}{3x+5y=-8}{4x+13y=2}

$$\left. \begin{array} { l } { 3 x + 5 y = - 8 } \\ { 4 x + 13 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-6$$

$$y=2$$

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.

$$3x+5y=-8,4x+13y=2$$

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

$$3x+5y=-8$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y-8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y-8\right)$$

Whakareatia \frac{1}{3} ki te -5y-8.

$$x=-\frac{5}{3}y-\frac{8}{3}$$

Whakakapia te \frac{-5y-8}{3} mō te x ki tērā atu whārite, 4x+13y=2.

$$4\left(-\frac{5}{3}y-\frac{8}{3}\right)+13y=2$$

Whakareatia 4 ki te \frac{-5y-8}{3}.

$$-\frac{20}{3}y-\frac{32}{3}+13y=2$$

Tāpiri -\frac{20y}{3} ki te 13y.

$$\frac{19}{3}y-\frac{32}{3}=2$$

Me tāpiri \frac{32}{3} ki ngā taha e rua o te whārite.

$$\frac{19}{3}y=\frac{38}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{3}, 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{5}{3}y-\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\times 2-\frac{8}{3}$$

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

$$x=\frac{-10-8}{3}$$

Tāpiri -\frac{8}{3} ki te -\frac{10}{3} 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.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=2$$

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.

$$3x+5y=-8,4x+13y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\4&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\4&13\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\4&13\end{matrix}\right))\left(\begin{matrix}3&5\\4&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\4&13\end{matrix}\right))\left(\begin{matrix}-8\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\4&13\end{matrix}\right))\left(\begin{matrix}-8\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\4&13\end{matrix}\right))\left(\begin{matrix}-8\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{3\times 13-5\times 4}&-\frac{5}{3\times 13-5\times 4}\\-\frac{4}{3\times 13-5\times 4}&\frac{3}{3\times 13-5\times 4}\end{matrix}\right)\left(\begin{matrix}-8\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{19}&-\frac{5}{19}\\-\frac{4}{19}&\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}-8\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{19}\left(-8\right)-\frac{5}{19}\times 2\\-\frac{4}{19}\left(-8\right)+\frac{3}{19}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-6,y=2$$

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.

$$3x+5y=-8,4x+13y=2$$

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.

$$4\times 3x+4\times 5y=4\left(-8\right),3\times 4x+3\times 13y=3\times 2$$

Whakarūnātia.

$$12x+20y=-32,12x+39y=6$$

Me tango 12x+39y=6 mai i 12x+20y=-32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+20y-39y=-32-6$$

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.

$$20y-39y=-32-6$$

Tāpiri 20y ki te -39y.

$$-19y=-32-6$$

Tāpiri -32 ki te -6.

$$-19y=-38$$

Whakawehea ngā taha e rua ki te -19.

$$y=2$$

Whakaurua te 2 mō y ki 4x+13y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+13\times 2=2$$

Whakareatia 13 ki te 2.

$$4x+26=2$$

Me tango 26 mai i ngā taha e rua o te whārite.

$$4x=-24$$

Whakawehea ngā taha e rua ki te 4.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=2$$