Whakaoti i te {l}{2x-y=2}{3x=2(5-y)}

$$\left\{ \begin{array} { l } { 2 x - y = 2 } \\ { 3 x = 2 ( 5 - y ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 5-y.

$$3x=10-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$3x+2y=10$$

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.

$$2x-y=2,3x+2y=10$$

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

$$2x-y=2$$

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

$$2x=y+2$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{y}{2}+1 mō te x ki tērā atu whārite, 3x+2y=10.

$$3\left(\frac{1}{2}y+1\right)+2y=10$$

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

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

Tāpiri \frac{3y}{2} ki te 2y.

$$\frac{7}{2}y+3=10$$

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

$$\frac{7}{2}y=7$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

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

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

Whakareatia \frac{1}{2} ki te 2.

$$x=1+1$$

Tāpiri 1 ki te 1.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 5-y.

$$3x=10-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$3x+2y=10$$

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

$$2x-y=2,3x+2y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-1\\3&2\end{matrix}\right))\left(\begin{matrix}2&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&2\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&2\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&2\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-\left(-3\right)}&-\frac{-1}{2\times 2-\left(-3\right)}\\-\frac{3}{2\times 2-\left(-3\right)}&\frac{2}{2\times 2-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}2\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{1}{7}\\-\frac{3}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}2\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 2+\frac{1}{7}\times 10\\-\frac{3}{7}\times 2+\frac{2}{7}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 5-y.

$$3x=10-2y$$

Me tāpiri te 2y ki ngā taha e rua.

$$3x+2y=10$$

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.

$$2x-y=2,3x+2y=10$$

Kia ōrite ai a 2x me 3x, 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 2.

$$3\times 2x+3\left(-1\right)y=3\times 2,2\times 3x+2\times 2y=2\times 10$$

Whakarūnātia.

$$6x-3y=6,6x+4y=20$$

Me tango 6x+4y=20 mai i 6x-3y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-3y-4y=6-20$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-4y=6-20$$

Tāpiri -3y ki te -4y.

$$-7y=6-20$$

Tāpiri 6 ki te -20.

$$-7y=-14$$

Whakawehea ngā taha e rua ki te -7.

$$y=2$$

Whakaurua te 2 mō y ki 3x+2y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 2=10$$

Whakareatia 2 ki te 2.

$$3x+4=10$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=2$$