Whakaoti i te {l}{x+y=45}{18x+120y=6000}

$$\left\{ \begin{array} { l } { x + y = 45 } \\ { 18 x + 120 y = 6000 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{100}{17} = -5\frac{15}{17} \approx -5.882352941$$

$$y = \frac{865}{17} = 50\frac{15}{17} \approx 50.882352941$$

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.

$$x+y=45,18x+120y=6000$$

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

$$x+y=45$$

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

$$x=-y+45$$

Whakakapia te -y+45 mō te x ki tērā atu whārite, 18x+120y=6000.

$$18\left(-y+45\right)+120y=6000$$

Whakareatia 18 ki te -y+45.

$$-18y+810+120y=6000$$

Tāpiri -18y ki te 120y.

$$102y+810=6000$$

Me tango 810 mai i ngā taha e rua o te whārite.

$$102y=5190$$

Whakawehea ngā taha e rua ki te 102.

$$y=\frac{865}{17}$$

Whakaurua te \frac{865}{17} mō y ki x=-y+45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{865}{17}+45$$

Tāpiri 45 ki te -\frac{865}{17}.

$$x=-\frac{100}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{100}{17},y=\frac{865}{17}$$

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.

$$x+y=45,18x+120y=6000$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\18&120\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\6000\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\18&120\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\18&120\end{matrix}\right))\left(\begin{matrix}1&1\\18&120\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&120\end{matrix}\right))\left(\begin{matrix}45\\6000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&120\end{matrix}\right))\left(\begin{matrix}45\\6000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&120\end{matrix}\right))\left(\begin{matrix}45\\6000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{120}{120-18}&-\frac{1}{120-18}\\-\frac{18}{120-18}&\frac{1}{120-18}\end{matrix}\right)\left(\begin{matrix}45\\6000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{17}&-\frac{1}{102}\\-\frac{3}{17}&\frac{1}{102}\end{matrix}\right)\left(\begin{matrix}45\\6000\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{17}\times 45-\frac{1}{102}\times 6000\\-\frac{3}{17}\times 45+\frac{1}{102}\times 6000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{100}{17}\\\frac{865}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{100}{17},y=\frac{865}{17}$$

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.

$$x+y=45,18x+120y=6000$$

Kia ōrite ai a x me 18x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 18 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$18x+18y=18\times 45,18x+120y=6000$$

Whakarūnātia.

$$18x+18y=810,18x+120y=6000$$

Me tango 18x+120y=6000 mai i 18x+18y=810 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+18y-120y=810-6000$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-120y=810-6000$$

Tāpiri 18y ki te -120y.

$$-102y=810-6000$$

Tāpiri 810 ki te -6000.

$$-102y=-5190$$

Whakawehea ngā taha e rua ki te -102.

$$y=\frac{865}{17}$$

Whakaurua te \frac{865}{17} mō y ki 18x+120y=6000. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$18x+120\times \frac{865}{17}=6000$$

Whakareatia 120 ki te \frac{865}{17}.

$$18x+\frac{103800}{17}=6000$$

Me tango \frac{103800}{17} mai i ngā taha e rua o te whārite.

$$18x=-\frac{1800}{17}$$

Whakawehea ngā taha e rua ki te 18.

$$x=-\frac{100}{17}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{100}{17},y=\frac{865}{17}$$