Whakaoti i te {l}{2x+y=40}{1x+3y=40}

$$\left.\begin{array} { l } { 2 x + y = 40 } \\ { 1 x + 3 y = 40 } \end{array} \right\}$$

Whakaoti mō x, y

$$x=16$$

$$y=8$$

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.

$$2x+y=40,x+3y=40$$

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

$$2x+y=40$$

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

$$2x=-y+40$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+40\right)$$

Whakareatia \frac{1}{2} ki te -y+40.

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

Whakakapia te -\frac{y}{2}+20 mō te x ki tērā atu whārite, x+3y=40.

$$-\frac{1}{2}y+20+3y=40$$

Tāpiri -\frac{y}{2} ki te 3y.

$$\frac{5}{2}y+20=40$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$\frac{5}{2}y=20$$

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

$$y=8$$

Whakaurua te 8 mō y ki x=-\frac{1}{2}y+20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 8+20$$

Whakareatia -\frac{1}{2} ki te 8.

$$x=-4+20$$

Tāpiri 20 ki te -4.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=8$$

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.

$$2x+y=40,x+3y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\40\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\1&3\end{matrix}\right))\left(\begin{matrix}2&1\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&3\end{matrix}\right))\left(\begin{matrix}40\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&3\end{matrix}\right))\left(\begin{matrix}40\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&3\end{matrix}\right))\left(\begin{matrix}40\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-1}&-\frac{1}{2\times 3-1}\\-\frac{1}{2\times 3-1}&\frac{2}{2\times 3-1}\end{matrix}\right)\left(\begin{matrix}40\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&-\frac{1}{5}\\-\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}40\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 40-\frac{1}{5}\times 40\\-\frac{1}{5}\times 40+\frac{2}{5}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=16,y=8$$

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.

$$2x+y=40,x+3y=40$$

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.

$$2x+y=40,2x+2\times 3y=2\times 40$$

Whakarūnātia.

$$2x+y=40,2x+6y=80$$

Me tango 2x+6y=80 mai i 2x+y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y-6y=40-80$$

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.

$$y-6y=40-80$$

Tāpiri y ki te -6y.

$$-5y=40-80$$

Tāpiri 40 ki te -80.

$$-5y=-40$$

Whakawehea ngā taha e rua ki te -5.

$$y=8$$

Whakaurua te 8 mō y ki x+3y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times 8=40$$

Whakareatia 3 ki te 8.

$$x+24=40$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=8$$