Whakaoti i te x-3y=7text{and}3x+3y=9

$$x - 3 y = 7 \text { and } 3 x + 3 y = 9$$

Whakaoti mō x, y

$$x=4$$

$$y=-1$$

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.

$$x-3y=7,3x+3y=9$$

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

$$x-3y=7$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y+7$$

Whakakapia te 3y+7 mō te x ki tērā atu whārite, 3x+3y=9.

$$3\left(3y+7\right)+3y=9$$

Whakareatia 3 ki te 3y+7.

$$9y+21+3y=9$$

Tāpiri 9y ki te 3y.

$$12y+21=9$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$12y=-12$$

Whakawehea ngā taha e rua ki te 12.

$$y=-1$$

Whakaurua te -1 mō y ki x=3y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-1\right)+7$$

Whakareatia 3 ki te -1.

$$x=-3+7$$

Tāpiri 7 ki te -3.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-1$$

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.

$$x-3y=7,3x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\3&3\end{matrix}\right))\left(\begin{matrix}1&-3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&3\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&3\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&3\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-3\times 3\right)}&-\frac{-3}{3-\left(-3\times 3\right)}\\-\frac{3}{3-\left(-3\times 3\right)}&\frac{1}{3-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{4}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 7+\frac{1}{4}\times 9\\-\frac{1}{4}\times 7+\frac{1}{12}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-1$$

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.

$$x-3y=7,3x+3y=9$$

Kia ōrite ai a x 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 1.

$$3x+3\left(-3\right)y=3\times 7,3x+3y=9$$

Whakarūnātia.

$$3x-9y=21,3x+3y=9$$

Me tango 3x+3y=9 mai i 3x-9y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-9y-3y=21-9$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-3y=21-9$$

Tāpiri -9y ki te -3y.

$$-12y=21-9$$

Tāpiri 21 ki te -9.

$$-12y=12$$

Whakawehea ngā taha e rua ki te -12.

$$y=-1$$

Whakaurua te -1 mō y ki 3x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+3\left(-1\right)=9$$

Whakareatia 3 ki te -1.

$$3x-3=9$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-1$$