Whakaoti i te {l}{2x+3y=8}{3x+3y=9}

$$\left. \begin{array} { l } { 2 x + 3 y = 8 } \\ { 3 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$2x+3y=8,3x+3y=9$$

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

$$2x+3y=8$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+8$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+8\right)$$

Whakareatia \frac{1}{2} ki te -3y+8.

$$x=-\frac{3}{2}y+4$$

Whakakapia te -\frac{3y}{2}+4 mō te x ki tērā atu whārite, 3x+3y=9.

$$3\left(-\frac{3}{2}y+4\right)+3y=9$$

Whakareatia 3 ki te -\frac{3y}{2}+4.

$$-\frac{9}{2}y+12+3y=9$$

Tāpiri -\frac{9y}{2} ki te 3y.

$$-\frac{3}{2}y+12=9$$

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

$$-\frac{3}{2}y=-3$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{3}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 2+4$$

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

$$x=-3+4$$

Tāpiri 4 ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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.

$$2x+3y=8,3x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\3&3\end{matrix}\right))\left(\begin{matrix}2&3\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&3\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-3\times 3}&-\frac{3}{2\times 3-3\times 3}\\-\frac{3}{2\times 3-3\times 3}&\frac{2}{2\times 3-3\times 3}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\1&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8+9\\8-\frac{2}{3}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$2x+3y=8,3x+3y=9$$

Me tango 3x+3y=9 mai i 2x+3y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-3x+3y-3y=8-9$$

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.

$$2x-3x=8-9$$

Tāpiri 2x ki te -3x.

$$-x=8-9$$

Tāpiri 8 ki te -9.

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

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

$$3+3y=9$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$3y=6$$

Whakawehea ngā taha e rua ki te 3.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$