Whakaoti i te {l}{3x+2y=3}{x-y=21}

$$\left. \begin{array} { l } { 3 x + 2 y = 3 } \\ { x - y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=-12$$

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.

$$3x+2y=3,x-y=21$$

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

$$3x+2y=3$$

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

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te -\frac{2y}{3}+1 mō te x ki tērā atu whārite, x-y=21.

$$-\frac{2}{3}y+1-y=21$$

Tāpiri -\frac{2y}{3} ki te -y.

$$-\frac{5}{3}y+1=21$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-\frac{5}{3}y=20$$

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.

$$y=-12$$

Whakaurua te -12 mō y ki x=-\frac{2}{3}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\left(-12\right)+1$$

Whakareatia -\frac{2}{3} ki te -12.

$$x=8+1$$

Tāpiri 1 ki te 8.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-12$$

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.

$$3x+2y=3,x-y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\21\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\1&-1\end{matrix}\right))\left(\begin{matrix}3&2\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\21\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-2}&-\frac{2}{3\left(-1\right)-2}\\-\frac{1}{3\left(-1\right)-2}&\frac{3}{3\left(-1\right)-2}\end{matrix}\right)\left(\begin{matrix}3\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{2}{5}\\\frac{1}{5}&-\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}3\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 3+\frac{2}{5}\times 21\\\frac{1}{5}\times 3-\frac{3}{5}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=-12$$

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.

$$3x+2y=3,x-y=21$$

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.

$$3x+2y=3,3x+3\left(-1\right)y=3\times 21$$

Whakarūnātia.

$$3x+2y=3,3x-3y=63$$

Me tango 3x-3y=63 mai i 3x+2y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+2y+3y=3-63$$

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.

$$2y+3y=3-63$$

Tāpiri 2y ki te 3y.

$$5y=3-63$$

Tāpiri 3 ki te -63.

$$5y=-60$$

Whakawehea ngā taha e rua ki te 5.

$$y=-12$$

Whakaurua te -12 mō y ki x-y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\left(-12\right)=21$$

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

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-12$$