Whakaoti i te {l}{36x-5y=7}{6x+3y=8}

$$\left. \begin{array} { l } { 36 x - 5 y = 7 } \\ { 6 x + 3 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{61}{138}\approx 0.442028986$$

$$y = \frac{41}{23} = 1\frac{18}{23} \approx 1.782608696$$

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.

$$36x-5y=7,6x+3y=8$$

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

$$36x-5y=7$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$36x=5y+7$$

Whakawehea ngā taha e rua ki te 36.

$$x=\frac{1}{36}\left(5y+7\right)$$

Whakareatia \frac{1}{36} ki te 5y+7.

$$x=\frac{5}{36}y+\frac{7}{36}$$

Whakakapia te \frac{5y+7}{36} mō te x ki tērā atu whārite, 6x+3y=8.

$$6\left(\frac{5}{36}y+\frac{7}{36}\right)+3y=8$$

Whakareatia 6 ki te \frac{5y+7}{36}.

$$\frac{5}{6}y+\frac{7}{6}+3y=8$$

Tāpiri \frac{5y}{6} ki te 3y.

$$\frac{23}{6}y+\frac{7}{6}=8$$

Me tango \frac{7}{6} mai i ngā taha e rua o te whārite.

$$\frac{23}{6}y=\frac{41}{6}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{41}{23}$$

Whakaurua te \frac{41}{23} mō y ki x=\frac{5}{36}y+\frac{7}{36}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{36}\times \frac{41}{23}+\frac{7}{36}$$

Whakareatia \frac{5}{36} ki te \frac{41}{23} 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.

$$x=\frac{205}{828}+\frac{7}{36}$$

Tāpiri \frac{7}{36} ki te \frac{205}{828} 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.

$$x=\frac{61}{138}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{61}{138},y=\frac{41}{23}$$

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.

$$36x-5y=7,6x+3y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}36&-5\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}36&-5\\6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}36&-5\\6&3\end{matrix}\right))\left(\begin{matrix}36&-5\\6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}36&-5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}36&-5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}36&-5\\6&3\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{36\times 3-\left(-5\times 6\right)}&-\frac{-5}{36\times 3-\left(-5\times 6\right)}\\-\frac{6}{36\times 3-\left(-5\times 6\right)}&\frac{36}{36\times 3-\left(-5\times 6\right)}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{46}&\frac{5}{138}\\-\frac{1}{23}&\frac{6}{23}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{46}\times 7+\frac{5}{138}\times 8\\-\frac{1}{23}\times 7+\frac{6}{23}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{61}{138}\\\frac{41}{23}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{61}{138},y=\frac{41}{23}$$

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.

$$36x-5y=7,6x+3y=8$$

Kia ōrite ai a 36x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 36.

$$6\times 36x+6\left(-5\right)y=6\times 7,36\times 6x+36\times 3y=36\times 8$$

Whakarūnātia.

$$216x-30y=42,216x+108y=288$$

Me tango 216x+108y=288 mai i 216x-30y=42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$216x-216x-30y-108y=42-288$$

Tāpiri 216x ki te -216x. Ka whakakore atu ngā kupu 216x me -216x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-30y-108y=42-288$$

Tāpiri -30y ki te -108y.

$$-138y=42-288$$

Tāpiri 42 ki te -288.

$$-138y=-246$$

Whakawehea ngā taha e rua ki te -138.

$$y=\frac{41}{23}$$

Whakaurua te \frac{41}{23} mō y ki 6x+3y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\times \frac{41}{23}=8$$

Whakareatia 3 ki te \frac{41}{23}.

$$6x+\frac{123}{23}=8$$

Me tango \frac{123}{23} mai i ngā taha e rua o te whārite.

$$6x=\frac{61}{23}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{61}{138}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{61}{138},y=\frac{41}{23}$$