Whakaoti i te {l}{3x+7y=6}{x+3y=12}

$$\left. \begin{array} { l } { 3 x + 7 y = 6 } \\ { x + 3 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-33$$

$$y=15$$

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.

$$3x+7y=6,x+3y=12$$

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

$$3x+7y=6$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$3x=-7y+6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-7y+6\right)$$

Whakareatia \frac{1}{3} ki te -7y+6.

$$x=-\frac{7}{3}y+2$$

Whakakapia te -\frac{7y}{3}+2 mō te x ki tērā atu whārite, x+3y=12.

$$-\frac{7}{3}y+2+3y=12$$

Tāpiri -\frac{7y}{3} ki te 3y.

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

Me tango 2 mai i ngā taha e rua o te whārite.

$$\frac{2}{3}y=10$$

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

$$y=15$$

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

$$x=-\frac{7}{3}\times 15+2$$

Whakareatia -\frac{7}{3} ki te 15.

$$x=-35+2$$

Tāpiri 2 ki te -35.

$$x=-33$$

Kua oti te pūnaha te whakatau.

$$x=-33,y=15$$

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.

$$3x+7y=6,x+3y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&7\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&7\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&7\\1&3\end{matrix}\right))\left(\begin{matrix}3&7\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\1&3\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\1&3\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\1&3\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-7}&-\frac{7}{3\times 3-7}\\-\frac{1}{3\times 3-7}&\frac{3}{3\times 3-7}\end{matrix}\right)\left(\begin{matrix}6\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}&-\frac{7}{2}\\-\frac{1}{2}&\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}6\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\times 6-\frac{7}{2}\times 12\\-\frac{1}{2}\times 6+\frac{3}{2}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-33\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-33,y=15$$

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.

$$3x+7y=6,x+3y=12$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+7y=6,3x+3\times 3y=3\times 12$$

Whakarūnātia.

$$3x+7y=6,3x+9y=36$$

Me tango 3x+9y=36 mai i 3x+7y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+7y-9y=6-36$$

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.

$$7y-9y=6-36$$

Tāpiri 7y ki te -9y.

$$-2y=6-36$$

Tāpiri 6 ki te -36.

$$-2y=-30$$

Whakawehea ngā taha e rua ki te -2.

$$y=15$$

Whakaurua te 15 mō y ki x+3y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times 15=12$$

Whakareatia 3 ki te 15.

$$x+45=12$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$x=-33$$

Kua oti te pūnaha te whakatau.

$$x=-33,y=15$$