Whakaoti i te {l}{3x-y=-9}{2x+3y=5}

$$\left. \begin{array} { l } { 3 x - y = - 9 } \\ { 2 x + 3 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=3$$

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.

$$3x-y=-9,2x+3y=5$$

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

$$3x-y=-9$$

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

$$3x=y-9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y-9\right)$$

Whakareatia \frac{1}{3} ki te y-9.

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

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

$$2\left(\frac{1}{3}y-3\right)+3y=5$$

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

$$\frac{2}{3}y-6+3y=5$$

Tāpiri \frac{2y}{3} ki te 3y.

$$\frac{11}{3}y-6=5$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$\frac{11}{3}y=11$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

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

$$x=\frac{1}{3}\times 3-3$$

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

$$x=1-3$$

Tāpiri -3 ki te 1.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=3$$

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.

$$3x-y=-9,2x+3y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}3&-1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-2\right)}&-\frac{-1}{3\times 3-\left(-2\right)}\\-\frac{2}{3\times 3-\left(-2\right)}&\frac{3}{3\times 3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&\frac{1}{11}\\-\frac{2}{11}&\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\left(-9\right)+\frac{1}{11}\times 5\\-\frac{2}{11}\left(-9\right)+\frac{3}{11}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=3$$

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.

$$3x-y=-9,2x+3y=5$$

Kia ōrite ai a 3x me 2x, 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.

$$2\times 3x+2\left(-1\right)y=2\left(-9\right),3\times 2x+3\times 3y=3\times 5$$

Whakarūnātia.

$$6x-2y=-18,6x+9y=15$$

Me tango 6x+9y=15 mai i 6x-2y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-2y-9y=-18-15$$

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.

$$-2y-9y=-18-15$$

Tāpiri -2y ki te -9y.

$$-11y=-18-15$$

Tāpiri -18 ki te -15.

$$-11y=-33$$

Whakawehea ngā taha e rua ki te -11.

$$y=3$$

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

$$2x+3\times 3=5$$

Whakareatia 3 ki te 3.

$$2x+9=5$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=3$$