Whakaoti i te {l}{3x-2y=5}{2x-3y=15}

$$\left. \begin{array} { l } { 3 x - 2 y = 5 } \\ { 2 x - 3 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-7$$

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.

$$3x-2y=5,2x-3y=15$$

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

$$3x-2y=5$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y+5$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te 2y+5.

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

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

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

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

$$\frac{4}{3}y+\frac{10}{3}-3y=15$$

Tāpiri \frac{4y}{3} ki te -3y.

$$-\frac{5}{3}y+\frac{10}{3}=15$$

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

$$-\frac{5}{3}y=\frac{35}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-7$$

Whakaurua te -7 mō y ki x=\frac{2}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\left(-7\right)+\frac{5}{3}$$

Whakareatia \frac{2}{3} ki te -7.

$$x=\frac{-14+5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{14}{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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

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

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.

$$3x-2y=5,2x-3y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right))\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\2&-3\end{matrix}\right))\left(\begin{matrix}5\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-\left(-2\times 2\right)}&-\frac{-2}{3\left(-3\right)-\left(-2\times 2\right)}\\-\frac{2}{3\left(-3\right)-\left(-2\times 2\right)}&\frac{3}{3\left(-3\right)-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}5\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&-\frac{2}{5}\\\frac{2}{5}&-\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}5\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 5-\frac{2}{5}\times 15\\\frac{2}{5}\times 5-\frac{3}{5}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x-2y=5,2x-3y=15$$

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.

$$2\times 3x+2\left(-2\right)y=2\times 5,3\times 2x+3\left(-3\right)y=3\times 15$$

Whakarūnātia.

$$6x-4y=10,6x-9y=45$$

Me tango 6x-9y=45 mai i 6x-4y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-4y+9y=10-45$$

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.

$$-4y+9y=10-45$$

Tāpiri -4y ki te 9y.

$$5y=10-45$$

Tāpiri 10 ki te -45.

$$5y=-35$$

Whakawehea ngā taha e rua ki te 5.

$$y=-7$$

Whakaurua te -7 mō y ki 2x-3y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-7\right)=15$$

Whakareatia -3 ki te -7.

$$2x+21=15$$

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

$$2x=-6$$

Whakawehea ngā taha e rua ki te 2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

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