Whakaoti i te {c}{-5x+3y=-8}{-x-3y=2}(1,-1)

$$\left. \begin{array} { c } { - 5 x + 3 y = - 8 } \\ { - x - 3 y = 2 } \end{array} \right. ( 1 , - 1 )$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

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

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

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

Me tango 3y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -5.

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

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

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

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

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

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

$$-\frac{3}{5}y-\frac{8}{5}-3y=2$$

Tāpiri -\frac{3y}{5} ki te -3y.

$$-\frac{18}{5}y-\frac{8}{5}=2$$

Me tāpiri \frac{8}{5} ki ngā taha e rua o te whārite.

$$-\frac{18}{5}y=\frac{18}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{18}{5}, 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{3}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

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

Tāpiri \frac{8}{5} ki te -\frac{3}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&-\frac{1}{6}\\\frac{1}{18}&-\frac{5}{18}\end{matrix}\right)\left(\begin{matrix}-8\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\left(-8\right)-\frac{1}{6}\times 2\\\frac{1}{18}\left(-8\right)-\frac{5}{18}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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.

$$-\left(-5\right)x-3y=-\left(-8\right),-5\left(-1\right)x-5\left(-3\right)y=-5\times 2$$

Whakarūnātia.

$$5x-3y=8,5x+15y=-10$$

Me tango 5x+15y=-10 mai i 5x-3y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-3y-15y=8+10$$

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.

$$-3y-15y=8+10$$

Tāpiri -3y ki te -15y.

$$-18y=8+10$$

Tāpiri 8 ki te 10.

$$-18y=18$$

Whakawehea ngā taha e rua ki te -18.

$$y=-1$$

Whakaurua te -1 mō y ki -x-3y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x-3\left(-1\right)=2$$

Whakareatia -3 ki te -1.

$$-x+3=2$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

Kua oti te pūnaha te whakatau.

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