Whakaoti i te {l}{x+3y=9}{x+y=5}

$$\left\{ \begin{array} { l } { x + 3 y = 9 } \\ { x + y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=2$$

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.

$$x+3y=9,x+y=5$$

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

$$x+3y=9$$

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

$$x=-3y+9$$

Whakakapia te -3y+9 mō te x ki tērā atu whārite, x+y=5.

$$-3y+9+y=5$$

Tāpiri -3y ki te y.

$$-2y+9=5$$

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

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

Whakaurua te 2 mō y ki x=-3y+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\times 2+9$$

Whakareatia -3 ki te 2.

$$x=-6+9$$

Tāpiri 9 ki te -6.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$

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.

$$x+3y=9,x+y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\5\end{matrix}\right)$$

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

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-3}&-\frac{3}{1-3}\\-\frac{1}{1-3}&\frac{1}{1-3}\end{matrix}\right)\left(\begin{matrix}9\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 9+\frac{3}{2}\times 5\\\frac{1}{2}\times 9-\frac{1}{2}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=2$$

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.

$$x+3y=9,x+y=5$$

Me tango x+y=5 mai i x+3y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+3y-y=9-5$$

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

$$3y-y=9-5$$

Tāpiri 3y ki te -y.

$$2y=9-5$$

Tāpiri 9 ki te -5.

$$2y=4$$

Whakawehea ngā taha e rua ki te 2.

$$y=2$$

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

$$x+2=5$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$