Whakaoti i te {l}{2x+2y=28}{x+3y=24}

$$\left. \begin{array} { l } { 2 x + 2 y = 28 } \\ { x + 3 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=5$$

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.

$$2x+2y=28,x+3y=24$$

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

$$2x+2y=28$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$2x=-2y+28$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-2y+28\right)$$

Whakareatia \frac{1}{2} ki te -2y+28.

$$x=-y+14$$

Whakakapia te -y+14 mō te x ki tērā atu whārite, x+3y=24.

$$-y+14+3y=24$$

Tāpiri -y ki te 3y.

$$2y+14=24$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$2y=10$$

Whakawehea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=-y+14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5+14$$

Tāpiri 14 ki te -5.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$

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.

$$2x+2y=28,x+3y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&2\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&2\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&2\\1&3\end{matrix}\right))\left(\begin{matrix}2&2\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\1&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\1&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\1&3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-2}&-\frac{2}{2\times 3-2}\\-\frac{1}{2\times 3-2}&\frac{2}{2\times 3-2}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&-\frac{1}{2}\\-\frac{1}{4}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 28-\frac{1}{2}\times 24\\-\frac{1}{4}\times 28+\frac{1}{2}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=5$$

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.

$$2x+2y=28,x+3y=24$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x+2y=28,2x+2\times 3y=2\times 24$$

Whakarūnātia.

$$2x+2y=28,2x+6y=48$$

Me tango 2x+6y=48 mai i 2x+2y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-6y=28-48$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-6y=28-48$$

Tāpiri 2y ki te -6y.

$$-4y=28-48$$

Tāpiri 28 ki te -48.

$$-4y=-20$$

Whakawehea ngā taha e rua ki te -4.

$$y=5$$

Whakaurua te 5 mō y ki x+3y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times 5=24$$

Whakareatia 3 ki te 5.

$$x+15=24$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$