Whakaoti i te {l}{27x/25+9/10y=1008}{11/10x+43/5y=1028}

$$\left. \begin{array} { l } { \frac { 27 x } { 25 } + \frac { 9 } { 10 } y = 1008 } \\ { \frac { 11 } { 10 } x + \frac { 43 } { 5 } y = 1028 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{430200}{461} = 933\frac{87}{461} \approx 933.188720174$$

$$y=\frac{80}{461}\approx 0.173535792$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 50, arā, te tauraro pātahi he tino iti rawa te kitea o 25,10.

$$2\times 27x+45y=50400$$

Whakareatia te 2 ki te 27, ka 54.

$$54x+45y=50400$$

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.

$$54x+45y=50400,\frac{11}{10}x+\frac{43}{5}y=1028$$

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

$$54x+45y=50400$$

Me tango 45y mai i ngā taha e rua o te whārite.

$$54x=-45y+50400$$

Whakawehea ngā taha e rua ki te 54.

$$x=\frac{1}{54}\left(-45y+50400\right)$$

Whakareatia \frac{1}{54} ki te -45y+50400.

$$x=-\frac{5}{6}y+\frac{2800}{3}$$

Whakakapia te -\frac{5y}{6}+\frac{2800}{3} mō te x ki tērā atu whārite, \frac{11}{10}x+\frac{43}{5}y=1028.

$$\frac{11}{10}\left(-\frac{5}{6}y+\frac{2800}{3}\right)+\frac{43}{5}y=1028$$

Whakareatia \frac{11}{10} ki te -\frac{5y}{6}+\frac{2800}{3}.

$$-\frac{11}{12}y+\frac{3080}{3}+\frac{43}{5}y=1028$$

Tāpiri -\frac{11y}{12} ki te \frac{43y}{5}.

$$\frac{461}{60}y+\frac{3080}{3}=1028$$

Me tango \frac{3080}{3} mai i ngā taha e rua o te whārite.

$$\frac{461}{60}y=\frac{4}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{461}{60}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{80}{461}$$

Whakaurua te \frac{80}{461} mō y ki x=-\frac{5}{6}y+\frac{2800}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{6}\times \frac{80}{461}+\frac{2800}{3}$$

Whakareatia -\frac{5}{6} ki te \frac{80}{461} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{200}{1383}+\frac{2800}{3}$$

Tāpiri \frac{2800}{3} ki te -\frac{200}{1383} 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.

$$x=\frac{430200}{461}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{430200}{461},y=\frac{80}{461}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 50, arā, te tauraro pātahi he tino iti rawa te kitea o 25,10.

$$2\times 27x+45y=50400$$

Whakareatia te 2 ki te 27, ka 54.

$$54x+45y=50400$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$54x+45y=50400,\frac{11}{10}x+\frac{43}{5}y=1028$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right).

$$inverse(\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right))\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right))\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right))\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}54&45\\\frac{11}{10}&\frac{43}{5}\end{matrix}\right))\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{43}{5}}{54\times \frac{43}{5}-45\times \frac{11}{10}}&-\frac{45}{54\times \frac{43}{5}-45\times \frac{11}{10}}\\-\frac{\frac{11}{10}}{54\times \frac{43}{5}-45\times \frac{11}{10}}&\frac{54}{54\times \frac{43}{5}-45\times \frac{11}{10}}\end{matrix}\right)\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{86}{4149}&-\frac{50}{461}\\-\frac{11}{4149}&\frac{60}{461}\end{matrix}\right)\left(\begin{matrix}50400\\1028\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{86}{4149}\times 50400-\frac{50}{461}\times 1028\\-\frac{11}{4149}\times 50400+\frac{60}{461}\times 1028\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{430200}{461}\\\frac{80}{461}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{430200}{461},y=\frac{80}{461}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 50, arā, te tauraro pātahi he tino iti rawa te kitea o 25,10.

$$2\times 27x+45y=50400$$

Whakareatia te 2 ki te 27, ka 54.

$$54x+45y=50400$$

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.

$$54x+45y=50400,\frac{11}{10}x+\frac{43}{5}y=1028$$

Kia ōrite ai a 54x me \frac{11x}{10}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{11}{10} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 54.

$$\frac{11}{10}\times 54x+\frac{11}{10}\times 45y=\frac{11}{10}\times 50400,54\times \frac{11}{10}x+54\times \frac{43}{5}y=54\times 1028$$

Whakarūnātia.

$$\frac{297}{5}x+\frac{99}{2}y=55440,\frac{297}{5}x+\frac{2322}{5}y=55512$$

Me tango \frac{297}{5}x+\frac{2322}{5}y=55512 mai i \frac{297}{5}x+\frac{99}{2}y=55440 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{297}{5}x-\frac{297}{5}x+\frac{99}{2}y-\frac{2322}{5}y=55440-55512$$

Tāpiri \frac{297x}{5} ki te -\frac{297x}{5}. Ka whakakore atu ngā kupu \frac{297x}{5} me -\frac{297x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{99}{2}y-\frac{2322}{5}y=55440-55512$$

Tāpiri \frac{99y}{2} ki te -\frac{2322y}{5}.

$$-\frac{4149}{10}y=55440-55512$$

Tāpiri 55440 ki te -55512.

$$-\frac{4149}{10}y=-72$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4149}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{80}{461}$$

Whakaurua te \frac{80}{461} mō y ki \frac{11}{10}x+\frac{43}{5}y=1028. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{11}{10}x+\frac{43}{5}\times \frac{80}{461}=1028$$

Whakareatia \frac{43}{5} ki te \frac{80}{461} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$\frac{11}{10}x+\frac{688}{461}=1028$$

Me tango \frac{688}{461} mai i ngā taha e rua o te whārite.

$$\frac{11}{10}x=\frac{473220}{461}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{430200}{461}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{430200}{461},y=\frac{80}{461}$$