Whakaoti i te {l}{2x+3y=5}{3x+2y=76}

$$\left. \begin{array} { l } { 2 x + 3 y = 5 } \\ { 3 x + 2 y = 76 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{218}{5} = 43\frac{3}{5} = 43.6$$

$$y = -\frac{137}{5} = -27\frac{2}{5} = -27.4$$

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.

$$2x+3y=5,3x+2y=76$$

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

$$2x+3y=5$$

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

$$2x=-3y+5$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+5\right)$$

Whakareatia \frac{1}{2} ki te -3y+5.

$$x=-\frac{3}{2}y+\frac{5}{2}$$

Whakakapia te \frac{-3y+5}{2} mō te x ki tērā atu whārite, 3x+2y=76.

$$3\left(-\frac{3}{2}y+\frac{5}{2}\right)+2y=76$$

Whakareatia 3 ki te \frac{-3y+5}{2}.

$$-\frac{9}{2}y+\frac{15}{2}+2y=76$$

Tāpiri -\frac{9y}{2} ki te 2y.

$$-\frac{5}{2}y+\frac{15}{2}=76$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-\frac{5}{2}y=\frac{137}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{137}{5}$$

Whakaurua te -\frac{137}{5} mō y ki x=-\frac{3}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\left(-\frac{137}{5}\right)+\frac{5}{2}$$

Whakareatia -\frac{3}{2} ki te -\frac{137}{5} 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.

$$x=\frac{411}{10}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{411}{10} 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.

$$x=\frac{218}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{218}{5},y=-\frac{137}{5}$$

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.

$$2x+3y=5,3x+2y=76$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\76\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\76\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\times 3}&-\frac{3}{2\times 2-3\times 3}\\-\frac{3}{2\times 2-3\times 3}&\frac{2}{2\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}5\\76\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{3}{5}\\\frac{3}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}5\\76\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\times 5+\frac{3}{5}\times 76\\\frac{3}{5}\times 5-\frac{2}{5}\times 76\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{218}{5}\\-\frac{137}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{218}{5},y=-\frac{137}{5}$$

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.

$$2x+3y=5,3x+2y=76$$

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.

$$3\times 2x+3\times 3y=3\times 5,2\times 3x+2\times 2y=2\times 76$$

Whakarūnātia.

$$6x+9y=15,6x+4y=152$$

Me tango 6x+4y=152 mai i 6x+9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+9y-4y=15-152$$

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.

$$9y-4y=15-152$$

Tāpiri 9y ki te -4y.

$$5y=15-152$$

Tāpiri 15 ki te -152.

$$5y=-137$$

Whakawehea ngā taha e rua ki te 5.

$$y=-\frac{137}{5}$$

Whakaurua te -\frac{137}{5} mō y ki 3x+2y=76. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-\frac{137}{5}\right)=76$$

Whakareatia 2 ki te -\frac{137}{5}.

$$3x-\frac{274}{5}=76$$

Me tāpiri \frac{274}{5} ki ngā taha e rua o te whārite.

$$3x=\frac{654}{5}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{218}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{218}{5},y=-\frac{137}{5}$$