Whakaoti i te {l}{x-y=12}{x+y=3}

$$\left\{ \begin{array} { l } { x - y = 12 } \\ { x + y = 3 } \end{array} \right\}$$

Whakaoti mō x, y

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$y = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

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.

$$x-y=12,x+y=3$$

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

$$x-y=12$$

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

$$x=y+12$$

Whakakapia te y+12 mō te x ki tērā atu whārite, x+y=3.

$$y+12+y=3$$

Tāpiri y ki te y.

$$2y+12=3$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$2y=-9$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{9}{2}$$

Whakaurua te -\frac{9}{2} mō y ki x=y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{2}+12$$

Tāpiri 12 ki te -\frac{9}{2}.

$$x=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{2},y=-\frac{9}{2}$$

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.

$$x-y=12,x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}12\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}12\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}12\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-1\right)}&-\frac{-1}{1-\left(-1\right)}\\-\frac{1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}12\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}12\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 12+\frac{1}{2}\times 3\\-\frac{1}{2}\times 12+\frac{1}{2}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2}\\-\frac{9}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{2},y=-\frac{9}{2}$$

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.

$$x-y=12,x+y=3$$

Me tango x+y=3 mai i x-y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-y-y=12-3$$

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.

$$-y-y=12-3$$

Tāpiri -y ki te -y.

$$-2y=12-3$$

Tāpiri 12 ki te -3.

$$-2y=9$$

Whakawehea ngā taha e rua ki te -2.

$$y=-\frac{9}{2}$$

Whakaurua te -\frac{9}{2} 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.

$$x-\frac{9}{2}=3$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{2},y=-\frac{9}{2}$$