Whakaoti i te {l}{3y+x=31}{2y+3x=44}

$$\left. \begin{array} { l } { 3 y + x = 31 } \\ { 2 y + 3 x = 44 } \end{array} \right.$$

Whakaoti mō y, x

$$x=10$$

$$y=7$$

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.

$$3y+x=31,2y+3x=44$$

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

$$3y+x=31$$

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

$$3y=-x+31$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(-x+31\right)$$

Whakareatia \frac{1}{3} ki te -x+31.

$$y=-\frac{1}{3}x+\frac{31}{3}$$

Whakakapia te \frac{-x+31}{3} mō te y ki tērā atu whārite, 2y+3x=44.

$$2\left(-\frac{1}{3}x+\frac{31}{3}\right)+3x=44$$

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

$$-\frac{2}{3}x+\frac{62}{3}+3x=44$$

Tāpiri -\frac{2x}{3} ki te 3x.

$$\frac{7}{3}x+\frac{62}{3}=44$$

Me tango \frac{62}{3} mai i ngā taha e rua o te whārite.

$$\frac{7}{3}x=\frac{70}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=10$$

Whakaurua te 10 mō x ki y=-\frac{1}{3}x+\frac{31}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{1}{3}\times 10+\frac{31}{3}$$

Whakareatia -\frac{1}{3} ki te 10.

$$y=\frac{-10+31}{3}$$

Tāpiri \frac{31}{3} ki te -\frac{10}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=7$$

Kua oti te pūnaha te whakatau.

$$y=7,x=10$$

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.

$$3y+x=31,2y+3x=44$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}31\\44\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\2&3\end{matrix}\right))\left(\begin{matrix}3&1\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&3\end{matrix}\right))\left(\begin{matrix}31\\44\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&3\end{matrix}\right))\left(\begin{matrix}31\\44\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&3\end{matrix}\right))\left(\begin{matrix}31\\44\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-2}&-\frac{1}{3\times 3-2}\\-\frac{2}{3\times 3-2}&\frac{3}{3\times 3-2}\end{matrix}\right)\left(\begin{matrix}31\\44\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&-\frac{1}{7}\\-\frac{2}{7}&\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}31\\44\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\times 31-\frac{1}{7}\times 44\\-\frac{2}{7}\times 31+\frac{3}{7}\times 44\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=7,x=10$$

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.

$$3y+x=31,2y+3x=44$$

Kia ōrite ai a 3y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3y+2x=2\times 31,3\times 2y+3\times 3x=3\times 44$$

Whakarūnātia.

$$6y+2x=62,6y+9x=132$$

Me tango 6y+9x=132 mai i 6y+2x=62 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+2x-9x=62-132$$

Tāpiri 6y ki te -6y. Ka whakakore atu ngā kupu 6y me -6y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2x-9x=62-132$$

Tāpiri 2x ki te -9x.

$$-7x=62-132$$

Tāpiri 62 ki te -132.

$$-7x=-70$$

Whakawehea ngā taha e rua ki te -7.

$$x=10$$

Whakaurua te 10 mō x ki 2y+3x=44. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+3\times 10=44$$

Whakareatia 3 ki te 10.

$$2y+30=44$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$2y=14$$

Whakawehea ngā taha e rua ki te 2.

$$y=7$$

Kua oti te pūnaha te whakatau.

$$y=7,x=10$$