Whakaoti i te {l}{2x+3y=62}{5x+y=90}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 62 } \\ { 5 x + y = 90 } \end{array} \right.$$

Whakaoti mō x, y

$$x=16$$

$$y=10$$

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.

$$2x+3y=62,5x+y=90$$

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

$$2x+3y=62$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+62$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+62\right)$$

Whakareatia \frac{1}{2} ki te -3y+62.

$$x=-\frac{3}{2}y+31$$

Whakakapia te -\frac{3y}{2}+31 mō te x ki tērā atu whārite, 5x+y=90.

$$5\left(-\frac{3}{2}y+31\right)+y=90$$

Whakareatia 5 ki te -\frac{3y}{2}+31.

$$-\frac{15}{2}y+155+y=90$$

Tāpiri -\frac{15y}{2} ki te y.

$$-\frac{13}{2}y+155=90$$

Me tango 155 mai i ngā taha e rua o te whārite.

$$-\frac{13}{2}y=-65$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki x=-\frac{3}{2}y+31. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 10+31$$

Whakareatia -\frac{3}{2} ki te 10.

$$x=-15+31$$

Tāpiri 31 ki te -15.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=10$$

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.

$$2x+3y=62,5x+y=90$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}62\\90\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\5&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\5&1\end{matrix}\right))\left(\begin{matrix}2&3\\5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&1\end{matrix}\right))\left(\begin{matrix}62\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&1\end{matrix}\right))\left(\begin{matrix}62\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&1\end{matrix}\right))\left(\begin{matrix}62\\90\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-3\times 5}&-\frac{3}{2-3\times 5}\\-\frac{5}{2-3\times 5}&\frac{2}{2-3\times 5}\end{matrix}\right)\left(\begin{matrix}62\\90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&\frac{3}{13}\\\frac{5}{13}&-\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}62\\90\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\times 62+\frac{3}{13}\times 90\\\frac{5}{13}\times 62-\frac{2}{13}\times 90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=16,y=10$$

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.

$$2x+3y=62,5x+y=90$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\times 3y=5\times 62,2\times 5x+2y=2\times 90$$

Whakarūnātia.

$$10x+15y=310,10x+2y=180$$

Me tango 10x+2y=180 mai i 10x+15y=310 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+15y-2y=310-180$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-2y=310-180$$

Tāpiri 15y ki te -2y.

$$13y=310-180$$

Tāpiri 310 ki te -180.

$$13y=130$$

Whakawehea ngā taha e rua ki te 13.

$$y=10$$

Whakaurua te 10 mō y ki 5x+y=90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+10=90$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$5x=80$$

Whakawehea ngā taha e rua ki te 5.

$$x=16$$

Kua oti te pūnaha te whakatau.

$$x=16,y=10$$