Whakaoti i te {l}{2x+3y=10}{3x+4y=15}

$$\left. \begin{array} { l } { 2 x + 3 y = 10 } \\ { 3 x + 4 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=0$$

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.

$$2x+3y=10,3x+4y=15$$

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

$$2x+3y=10$$

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

$$2x=-3y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+10\right)$$

Whakareatia \frac{1}{2} ki te -3y+10.

$$x=-\frac{3}{2}y+5$$

Whakakapia te -\frac{3y}{2}+5 mō te x ki tērā atu whārite, 3x+4y=15.

$$3\left(-\frac{3}{2}y+5\right)+4y=15$$

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

$$-\frac{9}{2}y+15+4y=15$$

Tāpiri -\frac{9y}{2} ki te 4y.

$$-\frac{1}{2}y+15=15$$

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

$$-\frac{1}{2}y=0$$

Me whakarea ngā taha e rua ki te -2.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{3}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$

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.

$$2x+3y=10,3x+4y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}2&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}10\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}10\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&4\end{matrix}\right))\left(\begin{matrix}10\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-3\times 3}&-\frac{3}{2\times 4-3\times 3}\\-\frac{3}{2\times 4-3\times 3}&\frac{2}{2\times 4-3\times 3}\end{matrix}\right)\left(\begin{matrix}10\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4&3\\3&-2\end{matrix}\right)\left(\begin{matrix}10\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\times 10+3\times 15\\3\times 10-2\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=0$$

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.

$$2x+3y=10,3x+4y=15$$

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

$$3\times 2x+3\times 3y=3\times 10,2\times 3x+2\times 4y=2\times 15$$

Whakarūnātia.

$$6x+9y=30,6x+8y=30$$

Me tango 6x+8y=30 mai i 6x+9y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+9y-8y=30-30$$

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

$$9y-8y=30-30$$

Tāpiri 9y ki te -8y.

$$y=30-30$$

Tāpiri 30 ki te -30.

$$y=0$$

Whakaurua te 0 mō y ki 3x+4y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$