Whakaoti i te {l}{x+y=30}{20x+25y=640}

$$\left\{ \begin{array} { l } { x + y = 30 } \\ { 20 x + 25 y = 640 } \end{array} \right.$$

Whakaoti mō x, y

$$x=22$$

$$y=8$$

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.

$$x+y=30,20x+25y=640$$

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

$$x+y=30$$

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

$$x=-y+30$$

Whakakapia te -y+30 mō te x ki tērā atu whārite, 20x+25y=640.

$$20\left(-y+30\right)+25y=640$$

Whakareatia 20 ki te -y+30.

$$-20y+600+25y=640$$

Tāpiri -20y ki te 25y.

$$5y+600=640$$

Me tango 600 mai i ngā taha e rua o te whārite.

$$5y=40$$

Whakawehea ngā taha e rua ki te 5.

$$y=8$$

Whakaurua te 8 mō y ki x=-y+30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-8+30$$

Tāpiri 30 ki te -8.

$$x=22$$

Kua oti te pūnaha te whakatau.

$$x=22,y=8$$

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.

$$x+y=30,20x+25y=640$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\20&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\640\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\20&25\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\20&25\end{matrix}\right))\left(\begin{matrix}1&1\\20&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\20&25\end{matrix}\right))\left(\begin{matrix}30\\640\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\20&25\end{matrix}\right))\left(\begin{matrix}30\\640\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\20&25\end{matrix}\right))\left(\begin{matrix}30\\640\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{25-20}&-\frac{1}{25-20}\\-\frac{20}{25-20}&\frac{1}{25-20}\end{matrix}\right)\left(\begin{matrix}30\\640\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&-\frac{1}{5}\\-4&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}30\\640\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 30-\frac{1}{5}\times 640\\-4\times 30+\frac{1}{5}\times 640\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}22\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=22,y=8$$

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.

$$x+y=30,20x+25y=640$$

Kia ōrite ai a x me 20x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 20 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$20x+20y=20\times 30,20x+25y=640$$

Whakarūnātia.

$$20x+20y=600,20x+25y=640$$

Me tango 20x+25y=640 mai i 20x+20y=600 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+20y-25y=600-640$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-25y=600-640$$

Tāpiri 20y ki te -25y.

$$-5y=600-640$$

Tāpiri 600 ki te -640.

$$-5y=-40$$

Whakawehea ngā taha e rua ki te -5.

$$y=8$$

Whakaurua te 8 mō y ki 20x+25y=640. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$20x+25\times 8=640$$

Whakareatia 25 ki te 8.

$$20x+200=640$$

Me tango 200 mai i ngā taha e rua o te whārite.

$$20x=440$$

Whakawehea ngā taha e rua ki te 20.

$$x=22$$

Kua oti te pūnaha te whakatau.

$$x=22,y=8$$