Whakaoti i te {l}{2x+5y=130}{4x+3y=218}

$$\left. \begin{array} { l } { 2 x + 5 y = 130 } \\ { 4 x + 3 y = 218 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=6$$

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.

$$2x+5y=130,4x+3y=218$$

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

$$2x+5y=130$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y+130$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+130\right)$$

Whakareatia \frac{1}{2} ki te -5y+130.

$$x=-\frac{5}{2}y+65$$

Whakakapia te -\frac{5y}{2}+65 mō te x ki tērā atu whārite, 4x+3y=218.

$$4\left(-\frac{5}{2}y+65\right)+3y=218$$

Whakareatia 4 ki te -\frac{5y}{2}+65.

$$-10y+260+3y=218$$

Tāpiri -10y ki te 3y.

$$-7y+260=218$$

Me tango 260 mai i ngā taha e rua o te whārite.

$$-7y=-42$$

Whakawehea ngā taha e rua ki te -7.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{5}{2}y+65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\times 6+65$$

Whakareatia -\frac{5}{2} ki te 6.

$$x=-15+65$$

Tāpiri 65 ki te -15.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=6$$

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.

$$2x+5y=130,4x+3y=218$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}130\\218\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}2&5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-5\times 4}&-\frac{5}{2\times 3-5\times 4}\\-\frac{4}{2\times 3-5\times 4}&\frac{2}{2\times 3-5\times 4}\end{matrix}\right)\left(\begin{matrix}130\\218\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{14}&\frac{5}{14}\\\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}130\\218\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{14}\times 130+\frac{5}{14}\times 218\\\frac{2}{7}\times 130-\frac{1}{7}\times 218\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=6$$

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.

$$2x+5y=130,4x+3y=218$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\times 5y=4\times 130,2\times 4x+2\times 3y=2\times 218$$

Whakarūnātia.

$$8x+20y=520,8x+6y=436$$

Me tango 8x+6y=436 mai i 8x+20y=520 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+20y-6y=520-436$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-6y=520-436$$

Tāpiri 20y ki te -6y.

$$14y=520-436$$

Tāpiri 520 ki te -436.

$$14y=84$$

Whakawehea ngā taha e rua ki te 14.

$$y=6$$

Whakaurua te 6 mō y ki 4x+3y=218. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\times 6=218$$

Whakareatia 3 ki te 6.

$$4x+18=218$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$4x=200$$

Whakawehea ngā taha e rua ki te 4.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=6$$