Whakaoti i te {l}{x-3y=7}{3x-3y=15}

$$\left. \begin{array} { l } { x - 3 y = 7 } \\ { 3 x - 3 y = 15 } \end{array} \right.$$

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.

$$x-3y=7,3x-3y=15$$

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

$$x-3y=7$$

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=15.

$$3\left(3y+7\right)-3y=15$$

Whakareatia 3 ki te 3y+7.

$$9y+21-3y=15$$

Tāpiri 9y ki te -3y.

$$6y+21=15$$

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

$$6y=-6$$

Whakawehea ngā taha e rua ki te 6.

$$y=-1$$

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.

$$x-3y=7,3x-3y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right))\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right))\left(\begin{matrix}7\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right))\left(\begin{matrix}7\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\3&-3\end{matrix}\right))\left(\begin{matrix}7\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-\left(-3\times 3\right)}&-\frac{-3}{-3-\left(-3\times 3\right)}\\-\frac{3}{-3-\left(-3\times 3\right)}&\frac{1}{-3-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}7\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}7\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 7+\frac{1}{2}\times 15\\-\frac{1}{2}\times 7+\frac{1}{6}\times 15\end{matrix}\right)$$

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.

$$x-3y=7,3x-3y=15$$

Me tango 3x-3y=15 mai i x-3y=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-3x-3y+3y=7-15$$

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

$$x-3x=7-15$$

Tāpiri x ki te -3x.

$$-2x=7-15$$

Tāpiri 7 ki te -15.

$$-2x=-8$$

Whakawehea ngā taha e rua ki te -2.

$$x=4$$

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

$$3\times 4-3y=15$$

Whakareatia 3 ki te 4.

$$12-3y=15$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-3y=3$$

Whakawehea ngā taha e rua ki te -3.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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