Whakaoti i te {l}{x+y=111}{x-y=101}

$$\left. \begin{array} { l } { x + y = 111 } \\ { x - y = 101 } \end{array} \right.$$

Whakaoti mō x, y

$$x=106$$

$$y=5$$

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.

$$x+y=111,x-y=101$$

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

$$x+y=111$$

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

$$x=-y+111$$

Whakakapia te -y+111 mō te x ki tērā atu whārite, x-y=101.

$$-y+111-y=101$$

Tāpiri -y ki te -y.

$$-2y+111=101$$

Me tango 111 mai i ngā taha e rua o te whārite.

$$-2y=-10$$

Whakawehea ngā taha e rua ki te -2.

$$y=5$$

Whakaurua te 5 mō y ki x=-y+111. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5+111$$

Tāpiri 111 ki te -5.

$$x=106$$

Kua oti te pūnaha te whakatau.

$$x=106,y=5$$

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.

$$x+y=111,x-y=101$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}111\\101\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}111\\101\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}111\\101\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}111\\101\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-1}&-\frac{1}{-1-1}\\-\frac{1}{-1-1}&\frac{1}{-1-1}\end{matrix}\right)\left(\begin{matrix}111\\101\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}111\\101\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 111+\frac{1}{2}\times 101\\\frac{1}{2}\times 111-\frac{1}{2}\times 101\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}106\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=106,y=5$$

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.

$$x+y=111,x-y=101$$

Me tango x-y=101 mai i x+y=111 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+y=111-101$$

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.

$$y+y=111-101$$

Tāpiri y ki te y.

$$2y=111-101$$

Tāpiri 111 ki te -101.

$$2y=10$$

Whakawehea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x-y=101. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-5=101$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x=106$$

Kua oti te pūnaha te whakatau.

$$x=106,y=5$$