Whakaoti i te {c}{3.90x+y=359.70}{-1.80x-y=-131}

$$\left. \begin{array} { c } { 3.90 x + y = 359.70 } \\ { - 1.80 x - y = - 131 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{2287}{21} = 108\frac{19}{21} \approx 108.904761905$$

$$y = -\frac{2276}{35} = -65\frac{1}{35} \approx -65.028571429$$

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.

$$3.9x+y=359.7,-1.8x-y=-131$$

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

$$3.9x+y=359.7$$

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

$$3.9x=-y+359.7$$

Whakawehea ngā taha e rua o te whārite ki te 3.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{10}{39}\left(-y+359.7\right)$$

Whakareatia \frac{10}{39} ki te -y+359.7.

$$x=-\frac{10}{39}y+\frac{1199}{13}$$

Whakakapia te -\frac{10y}{39}+\frac{1199}{13} mō te x ki tērā atu whārite, -1.8x-y=-131.

$$-1.8\left(-\frac{10}{39}y+\frac{1199}{13}\right)-y=-131$$

Whakareatia -1.8 ki te -\frac{10y}{39}+\frac{1199}{13}.

$$\frac{6}{13}y-\frac{10791}{65}-y=-131$$

Tāpiri \frac{6y}{13} ki te -y.

$$-\frac{7}{13}y-\frac{10791}{65}=-131$$

Me tāpiri \frac{10791}{65} ki ngā taha e rua o te whārite.

$$-\frac{7}{13}y=\frac{2276}{65}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{13}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{2276}{35}$$

Whakaurua te -\frac{2276}{35} mō y ki x=-\frac{10}{39}y+\frac{1199}{13}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{10}{39}\left(-\frac{2276}{35}\right)+\frac{1199}{13}$$

Whakareatia -\frac{10}{39} ki te -\frac{2276}{35} 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.

$$x=\frac{4552}{273}+\frac{1199}{13}$$

Tāpiri \frac{1199}{13} ki te \frac{4552}{273} 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.

$$x=\frac{2287}{21}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2287}{21},y=-\frac{2276}{35}$$

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.

$$3.9x+y=359.7,-1.8x-y=-131$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right))\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right))\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right))\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3.9&1\\-1.8&-1\end{matrix}\right))\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3.9\left(-1\right)-\left(-1.8\right)}&-\frac{1}{3.9\left(-1\right)-\left(-1.8\right)}\\-\frac{-1.8}{3.9\left(-1\right)-\left(-1.8\right)}&\frac{3.9}{3.9\left(-1\right)-\left(-1.8\right)}\end{matrix}\right)\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{21}&\frac{10}{21}\\-\frac{6}{7}&-\frac{13}{7}\end{matrix}\right)\left(\begin{matrix}359.7\\-131\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{21}\times 359.7+\frac{10}{21}\left(-131\right)\\-\frac{6}{7}\times 359.7-\frac{13}{7}\left(-131\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2287}{21}\\-\frac{2276}{35}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2287}{21},y=-\frac{2276}{35}$$

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.

$$3.9x+y=359.7,-1.8x-y=-131$$

Kia ōrite ai a \frac{39x}{10} me -\frac{9x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1.8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.9.

$$-1.8\times 3.9x-1.8y=-1.8\times 359.7,3.9\left(-1.8\right)x+3.9\left(-1\right)y=3.9\left(-131\right)$$

Whakarūnātia.

$$-7.02x-1.8y=-647.46,-7.02x-3.9y=-510.9$$

Me tango -7.02x-3.9y=-510.9 mai i -7.02x-1.8y=-647.46 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-7.02x+7.02x-1.8y+3.9y=-647.46+510.9$$

Tāpiri -\frac{351x}{50} ki te \frac{351x}{50}. Ka whakakore atu ngā kupu -\frac{351x}{50} me \frac{351x}{50}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-1.8y+3.9y=-647.46+510.9$$

Tāpiri -\frac{9y}{5} ki te \frac{39y}{10}.

$$2.1y=-647.46+510.9$$

Tāpiri -647.46 ki te 510.9 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.

$$2.1y=-136.56$$

Whakawehea ngā taha e rua o te whārite ki te 2.1, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{2276}{35}$$

Whakaurua te -\frac{2276}{35} mō y ki -1.8x-y=-131. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-1.8x-\left(-\frac{2276}{35}\right)=-131$$

Me tango \frac{2276}{35} mai i ngā taha e rua o te whārite.

$$-1.8x=-\frac{6861}{35}$$

Whakawehea ngā taha e rua o te whārite ki te -1.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{2287}{21}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2287}{21},y=-\frac{2276}{35}$$