Whakaoti i te {l}{y=.5x+2}{x+3y=1}

$$\left. \begin{array} { l } { y = .5 x + 2 } \\ { x + 3 y = 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-2$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 0.5x mai i ngā taha e rua.

$$y-0.5x=2$$

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.

$$y-0.5x=2,3y+x=1$$

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

$$y-0.5x=2$$

Me tāpiri \frac{x}{2} ki ngā taha e rua o te whārite.

$$y=0.5x+2$$

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

$$3\left(0.5x+2\right)+x=1$$

Whakareatia 3 ki te \frac{x}{2}+2.

$$1.5x+6+x=1$$

Tāpiri \frac{3x}{2} ki te x.

$$2.5x+6=1$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$2.5x=-5$$

Whakawehea ngā taha e rua o te whārite ki te 2.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-2$$

Whakaurua te -2 mō x ki y=0.5x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=0.5\left(-2\right)+2$$

Whakareatia 0.5 ki te -2.

$$y=-1+2$$

Tāpiri 2 ki te -1.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 0.5x mai i ngā taha e rua.

$$y-0.5x=2$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y-0.5x=2,3y+x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right))\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-0.5\\3&1\end{matrix}\right))\left(\begin{matrix}2\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-0.5\times 3\right)}&-\frac{-0.5}{1-\left(-0.5\times 3\right)}\\-\frac{3}{1-\left(-0.5\times 3\right)}&\frac{1}{1-\left(-0.5\times 3\right)}\end{matrix}\right)\left(\begin{matrix}2\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0.4&0.2\\-1.2&0.4\end{matrix}\right)\left(\begin{matrix}2\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0.4\times 2+0.2\\-1.2\times 2+0.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=1,x=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 0.5x mai i ngā taha e rua.

$$y-0.5x=2$$

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.

$$y-0.5x=2,3y+x=1$$

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

$$3y+3\left(-0.5\right)x=3\times 2,3y+x=1$$

Whakarūnātia.

$$3y-1.5x=6,3y+x=1$$

Me tango 3y+x=1 mai i 3y-1.5x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y-1.5x-x=6-1$$

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.

$$-1.5x-x=6-1$$

Tāpiri -\frac{3x}{2} ki te -x.

$$-2.5x=6-1$$

Tāpiri 6 ki te -1.

$$-2.5x=5$$

Whakawehea ngā taha e rua o te whārite ki te -2.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-2$$

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

$$3y-2=1$$

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

$$3y=3$$

Whakawehea ngā taha e rua ki te 3.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-2$$