Whakaoti i te {l}{3x-2y=1}{x+y=12}

$$\left\{ \begin{array} { l } { 3 x - 2 y = 1 } \\ { x + y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=7$$

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.

$$3x-2y=1,x+y=12$$

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

$$3x-2y=1$$

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

$$3x=2y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{2}{3}y+\frac{1}{3}$$

Whakakapia te \frac{2y+1}{3} mō te x ki tērā atu whārite, x+y=12.

$$\frac{2}{3}y+\frac{1}{3}+y=12$$

Tāpiri \frac{2y}{3} ki te y.

$$\frac{5}{3}y+\frac{1}{3}=12$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{35}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{2}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times 7+\frac{1}{3}$$

Whakareatia \frac{2}{3} ki te 7.

$$x=\frac{14+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{14}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$

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.

$$3x-2y=1,x+y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\1&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{2}{5}\\-\frac{1}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}1\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}+\frac{2}{5}\times 12\\-\frac{1}{5}+\frac{3}{5}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=7$$

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.

$$3x-2y=1,x+y=12$$

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.

$$3x-2y=1,3x+3y=3\times 12$$

Whakarūnātia.

$$3x-2y=1,3x+3y=36$$

Me tango 3x+3y=36 mai i 3x-2y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-2y-3y=1-36$$

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.

$$-2y-3y=1-36$$

Tāpiri -2y ki te -3y.

$$-5y=1-36$$

Tāpiri 1 ki te -36.

$$-5y=-35$$

Whakawehea ngā taha e rua ki te -5.

$$y=7$$

Whakaurua te 7 mō y ki x+y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+7=12$$

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

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$