Whakaoti i te {l}{3x-2y=-10}{5x-11y=-9}

$$\left\{ \begin{array} { l } { 3 x - 2 y = - 10 } \\ { 5 x - 11 y = - 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-1$$

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.

$$3x-2y=-10,5x-11y=-9$$

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

$$3x-2y=-10$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y-10$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(2y-10\right)$$

Whakareatia \frac{1}{3} ki te -10+2y.

$$x=\frac{2}{3}y-\frac{10}{3}$$

Whakakapia te \frac{-10+2y}{3} mō te x ki tērā atu whārite, 5x-11y=-9.

$$5\left(\frac{2}{3}y-\frac{10}{3}\right)-11y=-9$$

Whakareatia 5 ki te \frac{-10+2y}{3}.

$$\frac{10}{3}y-\frac{50}{3}-11y=-9$$

Tāpiri \frac{10y}{3} ki te -11y.

$$-\frac{23}{3}y-\frac{50}{3}=-9$$

Me tāpiri \frac{50}{3} ki ngā taha e rua o te whārite.

$$-\frac{23}{3}y=\frac{23}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{23}{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{2}{3}y-\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-1\right)-\frac{10}{3}$$

Whakareatia \frac{2}{3} ki te -1.

$$x=\frac{-2-10}{3}$$

Tāpiri -\frac{10}{3} ki te -\frac{2}{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.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-1$$

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.

$$3x-2y=-10,5x-11y=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\5&-11\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right))\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right))\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right))\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\5&-11\end{matrix}\right))\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{3\left(-11\right)-\left(-2\times 5\right)}&-\frac{-2}{3\left(-11\right)-\left(-2\times 5\right)}\\-\frac{5}{3\left(-11\right)-\left(-2\times 5\right)}&\frac{3}{3\left(-11\right)-\left(-2\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{23}&-\frac{2}{23}\\\frac{5}{23}&-\frac{3}{23}\end{matrix}\right)\left(\begin{matrix}-10\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{23}\left(-10\right)-\frac{2}{23}\left(-9\right)\\\frac{5}{23}\left(-10\right)-\frac{3}{23}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-1$$

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.

$$3x-2y=-10,5x-11y=-9$$

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5\left(-2\right)y=5\left(-10\right),3\times 5x+3\left(-11\right)y=3\left(-9\right)$$

Whakarūnātia.

$$15x-10y=-50,15x-33y=-27$$

Me tango 15x-33y=-27 mai i 15x-10y=-50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-10y+33y=-50+27$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y+33y=-50+27$$

Tāpiri -10y ki te 33y.

$$23y=-50+27$$

Tāpiri -50 ki te 27.

$$23y=-23$$

Whakawehea ngā taha e rua ki te 23.

$$y=-1$$

Whakaurua te -1 mō y ki 5x-11y=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-11\left(-1\right)=-9$$

Whakareatia -11 ki te -1.

$$5x+11=-9$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$5x=-20$$

Whakawehea ngā taha e rua ki te 5.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-1$$