Whakaoti i te {r}{2x+y=10}{x+y=4}

$$\left. \begin{array} { r } { 2 x + y = 10 } \\ { x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-2$$

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.

$$2x+y=10,x+y=4$$

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

$$2x+y=10$$

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

$$2x=-y+10$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-\frac{1}{2}y+5$$

Whakakapia te -\frac{y}{2}+5 mō te x ki tērā atu whārite, x+y=4.

$$-\frac{1}{2}y+5+y=4$$

Tāpiri -\frac{y}{2} ki te y.

$$\frac{1}{2}y+5=4$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$\frac{1}{2}y=-1$$

Me whakarea ngā taha e rua ki te 2.

$$y=-2$$

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

$$x=-\frac{1}{2}\left(-2\right)+5$$

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

$$x=1+5$$

Tāpiri 5 ki te 1.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-2$$

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.

$$2x+y=10,x+y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\1&1\end{matrix}\right))\left(\begin{matrix}2&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&1\end{matrix}\right))\left(\begin{matrix}10\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&1\end{matrix}\right))\left(\begin{matrix}10\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&1\end{matrix}\right))\left(\begin{matrix}10\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-1}&-\frac{1}{2-1}\\-\frac{1}{2-1}&\frac{2}{2-1}\end{matrix}\right)\left(\begin{matrix}10\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-1\\-1&2\end{matrix}\right)\left(\begin{matrix}10\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10-4\\-10+2\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-2$$

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.

$$2x+y=10,x+y=4$$

Me tango x+y=4 mai i 2x+y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-x+y-y=10-4$$

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

$$2x-x=10-4$$

Tāpiri 2x ki te -x.

$$x=10-4$$

Tāpiri 10 ki te -4.

$$x=6$$

Whakaurua te 6 mō x ki x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6+y=4$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-2$$