Whakaoti i te {l}{62x+y=44}{34x-y=36}

$$\left. \begin{array} { l } { 62 x + y = 44 } \\ { 34 x - y = 36 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{6}\approx 0.833333333$$

$$y = -\frac{23}{3} = -7\frac{2}{3} \approx -7.666666667$$

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.

$$62x+y=44,34x-y=36$$

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

$$62x+y=44$$

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

$$62x=-y+44$$

Whakawehea ngā taha e rua ki te 62.

$$x=\frac{1}{62}\left(-y+44\right)$$

Whakareatia \frac{1}{62} ki te -y+44.

$$x=-\frac{1}{62}y+\frac{22}{31}$$

Whakakapia te -\frac{y}{62}+\frac{22}{31} mō te x ki tērā atu whārite, 34x-y=36.

$$34\left(-\frac{1}{62}y+\frac{22}{31}\right)-y=36$$

Whakareatia 34 ki te -\frac{y}{62}+\frac{22}{31}.

$$-\frac{17}{31}y+\frac{748}{31}-y=36$$

Tāpiri -\frac{17y}{31} ki te -y.

$$-\frac{48}{31}y+\frac{748}{31}=36$$

Me tango \frac{748}{31} mai i ngā taha e rua o te whārite.

$$-\frac{48}{31}y=\frac{368}{31}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{48}{31}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{23}{3}$$

Whakaurua te -\frac{23}{3} mō y ki x=-\frac{1}{62}y+\frac{22}{31}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{62}\left(-\frac{23}{3}\right)+\frac{22}{31}$$

Whakareatia -\frac{1}{62} ki te -\frac{23}{3} 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.

$$x=\frac{23}{186}+\frac{22}{31}$$

Tāpiri \frac{22}{31} ki te \frac{23}{186} 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.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=-\frac{23}{3}$$

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.

$$62x+y=44,34x-y=36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}62&1\\34&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}44\\36\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}62&1\\34&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}62&1\\34&-1\end{matrix}\right))\left(\begin{matrix}62&1\\34&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}62&1\\34&-1\end{matrix}\right))\left(\begin{matrix}44\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}62&1\\34&-1\end{matrix}\right))\left(\begin{matrix}44\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}62&1\\34&-1\end{matrix}\right))\left(\begin{matrix}44\\36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{62\left(-1\right)-34}&-\frac{1}{62\left(-1\right)-34}\\-\frac{34}{62\left(-1\right)-34}&\frac{62}{62\left(-1\right)-34}\end{matrix}\right)\left(\begin{matrix}44\\36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{96}&\frac{1}{96}\\\frac{17}{48}&-\frac{31}{48}\end{matrix}\right)\left(\begin{matrix}44\\36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{96}\times 44+\frac{1}{96}\times 36\\\frac{17}{48}\times 44-\frac{31}{48}\times 36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\\-\frac{23}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{6},y=-\frac{23}{3}$$

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.

$$62x+y=44,34x-y=36$$

Kia ōrite ai a 62x me 34x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 34 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 62.

$$34\times 62x+34y=34\times 44,62\times 34x+62\left(-1\right)y=62\times 36$$

Whakarūnātia.

$$2108x+34y=1496,2108x-62y=2232$$

Me tango 2108x-62y=2232 mai i 2108x+34y=1496 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2108x-2108x+34y+62y=1496-2232$$

Tāpiri 2108x ki te -2108x. Ka whakakore atu ngā kupu 2108x me -2108x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$34y+62y=1496-2232$$

Tāpiri 34y ki te 62y.

$$96y=1496-2232$$

Tāpiri 1496 ki te -2232.

$$96y=-736$$

Whakawehea ngā taha e rua ki te 96.

$$y=-\frac{23}{3}$$

Whakaurua te -\frac{23}{3} mō y ki 34x-y=36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$34x-\left(-\frac{23}{3}\right)=36$$

Me tango \frac{23}{3} mai i ngā taha e rua o te whārite.

$$34x=\frac{85}{3}$$

Whakawehea ngā taha e rua ki te 34.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=-\frac{23}{3}$$