Whakaoti i te {l}{3x-8y=-13}{5y+2x=-19}

$$\left\{ \begin{array} { l } { 3 x - 8 y = - 13 } \\ { 5 y + 2 x = - 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=-1$$

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.

$$3x-8y=-13,2x+5y=-19$$

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

$$3x-8y=-13$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$3x=8y-13$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(8y-13\right)$$

Whakareatia \frac{1}{3} ki te 8y-13.

$$x=\frac{8}{3}y-\frac{13}{3}$$

Whakakapia te \frac{8y-13}{3} mō te x ki tērā atu whārite, 2x+5y=-19.

$$2\left(\frac{8}{3}y-\frac{13}{3}\right)+5y=-19$$

Whakareatia 2 ki te \frac{8y-13}{3}.

$$\frac{16}{3}y-\frac{26}{3}+5y=-19$$

Tāpiri \frac{16y}{3} ki te 5y.

$$\frac{31}{3}y-\frac{26}{3}=-19$$

Me tāpiri \frac{26}{3} ki ngā taha e rua o te whārite.

$$\frac{31}{3}y=-\frac{31}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{31}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{8}{3}y-\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{3}\left(-1\right)-\frac{13}{3}$$

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

$$x=\frac{-8-13}{3}$$

Tāpiri -\frac{13}{3} ki te -\frac{8}{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.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-1$$

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.

$$3x-8y=-13,2x+5y=-19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-8\\2&5\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-8\\2&5\end{matrix}\right))\left(\begin{matrix}-13\\-19\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}&\frac{8}{31}\\-\frac{2}{31}&\frac{3}{31}\end{matrix}\right)\left(\begin{matrix}-13\\-19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}\left(-13\right)+\frac{8}{31}\left(-19\right)\\-\frac{2}{31}\left(-13\right)+\frac{3}{31}\left(-19\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=-1$$

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.

$$3x-8y=-13,2x+5y=-19$$

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.

$$2\times 3x+2\left(-8\right)y=2\left(-13\right),3\times 2x+3\times 5y=3\left(-19\right)$$

Whakarūnātia.

$$6x-16y=-26,6x+15y=-57$$

Me tango 6x+15y=-57 mai i 6x-16y=-26 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-16y-15y=-26+57$$

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.

$$-16y-15y=-26+57$$

Tāpiri -16y ki te -15y.

$$-31y=-26+57$$

Tāpiri -26 ki te 57.

$$-31y=31$$

Whakawehea ngā taha e rua ki te -31.

$$y=-1$$

Whakaurua te -1 mō y ki 2x+5y=-19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\left(-1\right)=-19$$

Whakareatia 5 ki te -1.

$$2x-5=-19$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$2x=-14$$

Whakawehea ngā taha e rua ki te 2.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-1$$