Whakaoti i te {l}{3x-5y=-18}{3x-2y=9}

$$\left. \begin{array} { l } { 3 x - 5 y = - 18 } \\ { 3 x - 2 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=9$$

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.

$$3x-5y=-18,3x-2y=9$$

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

$$3x-5y=-18$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y-18$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y-18\right)$$

Whakareatia \frac{1}{3} ki te 5y-18.

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

Whakakapia te \frac{5y}{3}-6 mō te x ki tērā atu whārite, 3x-2y=9.

$$3\left(\frac{5}{3}y-6\right)-2y=9$$

Whakareatia 3 ki te \frac{5y}{3}-6.

$$5y-18-2y=9$$

Tāpiri 5y ki te -2y.

$$3y-18=9$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$3y=27$$

Whakawehea ngā taha e rua ki te 3.

$$y=9$$

Whakaurua te 9 mō y ki x=\frac{5}{3}y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\times 9-6$$

Whakareatia \frac{5}{3} ki te 9.

$$x=15-6$$

Tāpiri -6 ki te 15.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=9$$

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.

$$3x-5y=-18,3x-2y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right))\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right))\left(\begin{matrix}-18\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right))\left(\begin{matrix}-18\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\3&-2\end{matrix}\right))\left(\begin{matrix}-18\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-5\times 3\right)}&-\frac{-5}{3\left(-2\right)-\left(-5\times 3\right)}\\-\frac{3}{3\left(-2\right)-\left(-5\times 3\right)}&\frac{3}{3\left(-2\right)-\left(-5\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-18\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}&\frac{5}{9}\\-\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-18\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}\left(-18\right)+\frac{5}{9}\times 9\\-\frac{1}{3}\left(-18\right)+\frac{1}{3}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=9$$

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.

$$3x-5y=-18,3x-2y=9$$

Me tango 3x-2y=9 mai i 3x-5y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-5y+2y=-18-9$$

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.

$$-5y+2y=-18-9$$

Tāpiri -5y ki te 2y.

$$-3y=-18-9$$

Tāpiri -18 ki te -9.

$$-3y=-27$$

Whakawehea ngā taha e rua ki te -3.

$$y=9$$

Whakaurua te 9 mō y ki 3x-2y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times 9=9$$

Whakareatia -2 ki te 9.

$$3x-18=9$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$3x=27$$

Whakawehea ngā taha e rua ki te 3.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=9$$