Whakaoti i te {l}{80x+160y=4}{5600x+5600y=5536}

$$\left. \begin{array} { l } { 80 x + 160 y = 4 } \\ { 5600 x + 5600 y = 5536 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{1349}{700} = 1\frac{649}{700} \approx 1.927142857$$

$$y=-\frac{657}{700}\approx -0.938571429$$

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.

$$80x+160y=4,5600x+5600y=5536$$

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

$$80x+160y=4$$

Me tango 160y mai i ngā taha e rua o te whārite.

$$80x=-160y+4$$

Whakawehea ngā taha e rua ki te 80.

$$x=\frac{1}{80}\left(-160y+4\right)$$

Whakareatia \frac{1}{80} ki te -160y+4.

$$x=-2y+\frac{1}{20}$$

Whakakapia te -2y+\frac{1}{20} mō te x ki tērā atu whārite, 5600x+5600y=5536.

$$5600\left(-2y+\frac{1}{20}\right)+5600y=5536$$

Whakareatia 5600 ki te -2y+\frac{1}{20}.

$$-11200y+280+5600y=5536$$

Tāpiri -11200y ki te 5600y.

$$-5600y+280=5536$$

Me tango 280 mai i ngā taha e rua o te whārite.

$$-5600y=5256$$

Whakawehea ngā taha e rua ki te -5600.

$$y=-\frac{657}{700}$$

Whakaurua te -\frac{657}{700} mō y ki x=-2y+\frac{1}{20}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-\frac{657}{700}\right)+\frac{1}{20}$$

Whakareatia -2 ki te -\frac{657}{700}.

$$x=\frac{657}{350}+\frac{1}{20}$$

Tāpiri \frac{1}{20} ki te \frac{657}{350} 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.

$$x=\frac{1349}{700}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1349}{700},y=-\frac{657}{700}$$

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.

$$80x+160y=4,5600x+5600y=5536$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\5536\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}80&160\\5600&5600\end{matrix}\right).

$$inverse(\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right))\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right))\left(\begin{matrix}4\\5536\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right))\left(\begin{matrix}4\\5536\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}80&160\\5600&5600\end{matrix}\right))\left(\begin{matrix}4\\5536\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5600}{80\times 5600-160\times 5600}&-\frac{160}{80\times 5600-160\times 5600}\\-\frac{5600}{80\times 5600-160\times 5600}&\frac{80}{80\times 5600-160\times 5600}\end{matrix}\right)\left(\begin{matrix}4\\5536\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}&\frac{1}{2800}\\\frac{1}{80}&-\frac{1}{5600}\end{matrix}\right)\left(\begin{matrix}4\\5536\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}\times 4+\frac{1}{2800}\times 5536\\\frac{1}{80}\times 4-\frac{1}{5600}\times 5536\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1349}{700}\\-\frac{657}{700}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1349}{700},y=-\frac{657}{700}$$

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.

$$80x+160y=4,5600x+5600y=5536$$

Kia ōrite ai a 80x me 5600x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5600 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 80.

$$5600\times 80x+5600\times 160y=5600\times 4,80\times 5600x+80\times 5600y=80\times 5536$$

Whakarūnātia.

$$448000x+896000y=22400,448000x+448000y=442880$$

Me tango 448000x+448000y=442880 mai i 448000x+896000y=22400 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$448000x-448000x+896000y-448000y=22400-442880$$

Tāpiri 448000x ki te -448000x. Ka whakakore atu ngā kupu 448000x me -448000x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$896000y-448000y=22400-442880$$

Tāpiri 896000y ki te -448000y.

$$448000y=22400-442880$$

Tāpiri 22400 ki te -442880.

$$448000y=-420480$$

Whakawehea ngā taha e rua ki te 448000.

$$y=-\frac{657}{700}$$

Whakaurua te -\frac{657}{700} mō y ki 5600x+5600y=5536. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5600x+5600\left(-\frac{657}{700}\right)=5536$$

Whakareatia 5600 ki te -\frac{657}{700}.

$$5600x-5256=5536$$

Me tāpiri 5256 ki ngā taha e rua o te whārite.

$$5600x=10792$$

Whakawehea ngā taha e rua ki te 5600.

$$x=\frac{1349}{700}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1349}{700},y=-\frac{657}{700}$$