Whakaoti i te {l}{3x+3y=12}{3x+2y=13}

$$\left. \begin{array} { l } { 3 x + 3 y = 12 } \\ { 3 x + 2 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-1$$

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.

$$3x+3y=12,3x+2y=13$$

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

$$3x+3y=12$$

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

$$3x=-3y+12$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-3y+12\right)$$

Whakareatia \frac{1}{3} ki te -3y+12.

$$x=-y+4$$

Whakakapia te -y+4 mō te x ki tērā atu whārite, 3x+2y=13.

$$3\left(-y+4\right)+2y=13$$

Whakareatia 3 ki te -y+4.

$$-3y+12+2y=13$$

Tāpiri -3y ki te 2y.

$$-y+12=13$$

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

$$-y=1$$

Whakawehea ngā taha e rua ki te -1.

$$y=-1$$

Whakaurua te -1 mō y ki x=-y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-1\right)+4$$

Whakareatia -1 ki te -1.

$$x=1+4$$

Tāpiri 4 ki te 1.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-1$$

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.

$$3x+3y=12,3x+2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\13\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&3\\3&2\end{matrix}\right))\left(\begin{matrix}3&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&3\\3&2\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&3\\3&2\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&3\\3&2\end{matrix}\right))\left(\begin{matrix}12\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-3\times 3}&-\frac{3}{3\times 2-3\times 3}\\-\frac{3}{3\times 2-3\times 3}&\frac{3}{3\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}12\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&1\\1&-1\end{matrix}\right)\left(\begin{matrix}12\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 12+13\\12-13\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-1$$

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.

$$3x+3y=12,3x+2y=13$$

Me tango 3x+2y=13 mai i 3x+3y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+3y-2y=12-13$$

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.

$$3y-2y=12-13$$

Tāpiri 3y ki te -2y.

$$y=12-13$$

Tāpiri 12 ki te -13.

$$y=-1$$

Whakaurua te -1 mō y ki 3x+2y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-1\right)=13$$

Whakareatia 2 ki te -1.

$$3x-2=13$$

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

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-1$$