Whakaoti i te {l}{x+y=67.56}{x-y=12.40}

$$\left\{ \begin{array} { l } { x + y = 67.56 } \\ { x - y = 12.40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=39.98$$

$$y=27.58$$

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.

$$x+y=67.56,x-y=12.4$$

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

$$x+y=67.56$$

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

$$x=-y+67.56$$

Whakakapia te -y+67.56 mō te x ki tērā atu whārite, x-y=12.4.

$$-y+67.56-y=12.4$$

Tāpiri -y ki te -y.

$$-2y+67.56=12.4$$

Me tango 67.56 mai i ngā taha e rua o te whārite.

$$-2y=-55.16$$

Whakawehea ngā taha e rua ki te -2.

$$y=27.58$$

Whakaurua te 27.58 mō y ki x=-y+67.56. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-27.58+67.56$$

Tāpiri 67.56 ki te -27.58 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=39.98$$

Kua oti te pūnaha te whakatau.

$$x=39.98,y=27.58$$

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.

$$x+y=67.56,x-y=12.4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-1}&-\frac{1}{-1-1}\\-\frac{1}{-1-1}&\frac{1}{-1-1}\end{matrix}\right)\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}67.56\\12.4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 67.56+\frac{1}{2}\times 12.4\\\frac{1}{2}\times 67.56-\frac{1}{2}\times 12.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1999}{50}\\\frac{1379}{50}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1999}{50},y=\frac{1379}{50}$$

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.

$$x+y=67.56,x-y=12.4$$

Me tango x-y=12.4 mai i x+y=67.56 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+y=67.56-12.4$$

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.

$$y+y=67.56-12.4$$

Tāpiri y ki te y.

$$2y=67.56-12.4$$

Tāpiri 67.56 ki te -12.4 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$2y=55.16$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1379}{50}$$

Whakaurua te \frac{1379}{50} mō y ki x-y=12.4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{1379}{50}=12.4$$

Whakareatia -1 ki te \frac{1379}{50}.

$$x-27.58=12.4$$

Me tāpiri 27.58 ki ngā taha e rua o te whārite.

$$x=39.98$$

Kua oti te pūnaha te whakatau.

$$x=39.98,y=\frac{1379}{50}$$