Whakaoti i te {l}{5x-3y=1800}{6x-4y=1600}

$$\left. \begin{array} { l } { 5 x - 3 y = 1800 } \\ { 6 x - 4 y = 1600 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1200$$

$$y=1400$$

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.

$$5x-3y=1800,6x-4y=1600$$

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

$$5x-3y=1800$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+1800$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+1800\right)$$

Whakareatia \frac{1}{5} ki te 1800+3y.

$$x=\frac{3}{5}y+360$$

Whakakapia te \frac{3y}{5}+360 mō te x ki tērā atu whārite, 6x-4y=1600.

$$6\left(\frac{3}{5}y+360\right)-4y=1600$$

Whakareatia 6 ki te \frac{3y}{5}+360.

$$\frac{18}{5}y+2160-4y=1600$$

Tāpiri \frac{18y}{5} ki te -4y.

$$-\frac{2}{5}y+2160=1600$$

Me tango 2160 mai i ngā taha e rua o te whārite.

$$-\frac{2}{5}y=-560$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1400$$

Whakaurua te 1400 mō y ki x=\frac{3}{5}y+360. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\times 1400+360$$

Whakareatia \frac{3}{5} ki te 1400.

$$x=840+360$$

Tāpiri 360 ki te 840.

$$x=1200$$

Kua oti te pūnaha te whakatau.

$$x=1200,y=1400$$

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.

$$5x-3y=1800,6x-4y=1600$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\6&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right))\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right))\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right))\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\6&-4\end{matrix}\right))\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5\left(-4\right)-\left(-3\times 6\right)}&-\frac{-3}{5\left(-4\right)-\left(-3\times 6\right)}\\-\frac{6}{5\left(-4\right)-\left(-3\times 6\right)}&\frac{5}{5\left(-4\right)-\left(-3\times 6\right)}\end{matrix}\right)\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{3}{2}\\3&-\frac{5}{2}\end{matrix}\right)\left(\begin{matrix}1800\\1600\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 1800-\frac{3}{2}\times 1600\\3\times 1800-\frac{5}{2}\times 1600\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1200\\1400\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1200,y=1400$$

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.

$$5x-3y=1800,6x-4y=1600$$

Kia ōrite ai a 5x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$6\times 5x+6\left(-3\right)y=6\times 1800,5\times 6x+5\left(-4\right)y=5\times 1600$$

Whakarūnātia.

$$30x-18y=10800,30x-20y=8000$$

Me tango 30x-20y=8000 mai i 30x-18y=10800 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30x-30x-18y+20y=10800-8000$$

Tāpiri 30x ki te -30x. Ka whakakore atu ngā kupu 30x me -30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-18y+20y=10800-8000$$

Tāpiri -18y ki te 20y.

$$2y=10800-8000$$

Tāpiri 10800 ki te -8000.

$$2y=2800$$

Whakawehea ngā taha e rua ki te 2.

$$y=1400$$

Whakaurua te 1400 mō y ki 6x-4y=1600. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-4\times 1400=1600$$

Whakareatia -4 ki te 1400.

$$6x-5600=1600$$

Me tāpiri 5600 ki ngā taha e rua o te whārite.

$$6x=7200$$

Whakawehea ngā taha e rua ki te 6.

$$x=1200$$

Kua oti te pūnaha te whakatau.

$$x=1200,y=1400$$