Whakaoti i te {l}{2x+y=12}{3x-2y=8}

$$\left. \begin{array} { l } { 2 x + y = 12 } \\ { 3 x - 2 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{32}{7} = 4\frac{4}{7} \approx 4.571428571$$

$$y = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

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.

$$2x+y=12,3x-2y=8$$

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

$$2x+y=12$$

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

$$2x=-y+12$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -y+12.

$$x=-\frac{1}{2}y+6$$

Whakakapia te -\frac{y}{2}+6 mō te x ki tērā atu whārite, 3x-2y=8.

$$3\left(-\frac{1}{2}y+6\right)-2y=8$$

Whakareatia 3 ki te -\frac{y}{2}+6.

$$-\frac{3}{2}y+18-2y=8$$

Tāpiri -\frac{3y}{2} ki te -2y.

$$-\frac{7}{2}y+18=8$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-\frac{7}{2}y=-10$$

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.

$$y=\frac{20}{7}$$

Whakaurua te \frac{20}{7} mō y ki x=-\frac{1}{2}y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{20}{7}+6$$

Whakareatia -\frac{1}{2} ki te \frac{20}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{10}{7}+6$$

Tāpiri 6 ki te -\frac{10}{7}.

$$x=\frac{32}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{32}{7},y=\frac{20}{7}$$

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.

$$2x+y=12,3x-2y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\3&-2\end{matrix}\right))\left(\begin{matrix}2&1\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&-2\end{matrix}\right))\left(\begin{matrix}12\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&-2\end{matrix}\right))\left(\begin{matrix}12\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&-2\end{matrix}\right))\left(\begin{matrix}12\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-3}&-\frac{1}{2\left(-2\right)-3}\\-\frac{3}{2\left(-2\right)-3}&\frac{2}{2\left(-2\right)-3}\end{matrix}\right)\left(\begin{matrix}12\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{1}{7}\\\frac{3}{7}&-\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}12\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 12+\frac{1}{7}\times 8\\\frac{3}{7}\times 12-\frac{2}{7}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{32}{7}\\\frac{20}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{32}{7},y=\frac{20}{7}$$

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.

$$2x+y=12,3x-2y=8$$

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.

$$3\times 2x+3y=3\times 12,2\times 3x+2\left(-2\right)y=2\times 8$$

Whakarūnātia.

$$6x+3y=36,6x-4y=16$$

Me tango 6x-4y=16 mai i 6x+3y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+3y+4y=36-16$$

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.

$$3y+4y=36-16$$

Tāpiri 3y ki te 4y.

$$7y=36-16$$

Tāpiri 36 ki te -16.

$$7y=20$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{20}{7}$$

Whakaurua te \frac{20}{7} mō y ki 3x-2y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times \frac{20}{7}=8$$

Whakareatia -2 ki te \frac{20}{7}.

$$3x-\frac{40}{7}=8$$

Me tāpiri \frac{40}{7} ki ngā taha e rua o te whārite.

$$3x=\frac{96}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{32}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{32}{7},y=\frac{20}{7}$$