Whakaoti i te {l}{x+y=3}{3x+2y=5}

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

Whakaoti mō x, y

$$x=-1$$

$$y=4$$

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.

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

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

$$x+y=3$$

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

$$x=-y+3$$

Whakakapia te -y+3 mō te x ki tērā atu whārite, 3x+2y=5.

$$3\left(-y+3\right)+2y=5$$

Whakareatia 3 ki te -y+3.

$$-3y+9+2y=5$$

Tāpiri -3y ki te 2y.

$$-y+9=5$$

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

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

Whakaurua te 4 mō y ki x=-y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4+3$$

Tāpiri 3 ki te -4.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\3&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3y=3\times 3,3x+2y=5$$

Whakarūnātia.

$$3x+3y=9,3x+2y=5$$

Me tango 3x+2y=5 mai i 3x+3y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+3y-2y=9-5$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-2y=9-5$$

Tāpiri 3y ki te -2y.

$$y=9-5$$

Tāpiri 9 ki te -5.

$$y=4$$

Whakaurua te 4 mō y ki 3x+2y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 4=5$$

Whakareatia 2 ki te 4.

$$3x+8=5$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$3x=-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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