Whakaoti i te {l}{6.3x-4.1y=-3.5}{-8.7x+3.8y=-8.2}

$$\left. \begin{array}{l}{ 6.3 x - 4.1 y = - 3.5 }\\{ - 8.7 x + 3.8 y = - 8.2 }\end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=7$$

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.

$$6.3x-4.1y=-3.5,-8.7x+3.8y=-8.2$$

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

$$6.3x-4.1y=-3.5$$

Me tāpiri \frac{41y}{10} ki ngā taha e rua o te whārite.

$$6.3x=4.1y-3.5$$

Whakawehea ngā taha e rua o te whārite ki te 6.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{10}{63}\left(4.1y-3.5\right)$$

Whakareatia \frac{10}{63} ki te \frac{41y}{10}-3.5.

$$x=\frac{41}{63}y-\frac{5}{9}$$

Whakakapia te \frac{41y}{63}-\frac{5}{9} mō te x ki tērā atu whārite, -8.7x+3.8y=-8.2.

$$-8.7\left(\frac{41}{63}y-\frac{5}{9}\right)+3.8y=-8.2$$

Whakareatia -8.7 ki te \frac{41y}{63}-\frac{5}{9}.

$$-\frac{1189}{210}y+\frac{29}{6}+3.8y=-8.2$$

Tāpiri -\frac{1189y}{210} ki te \frac{19y}{5}.

$$-\frac{391}{210}y+\frac{29}{6}=-8.2$$

Me tango \frac{29}{6} mai i ngā taha e rua o te whārite.

$$-\frac{391}{210}y=-\frac{391}{30}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{391}{210}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{41}{63}y-\frac{5}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{41}{63}\times 7-\frac{5}{9}$$

Whakareatia \frac{41}{63} ki te 7.

$$x=\frac{41-5}{9}$$

Tāpiri -\frac{5}{9} ki te \frac{41}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$

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.

$$6.3x-4.1y=-3.5,-8.7x+3.8y=-8.2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right).

$$inverse(\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right))\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right))\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right))\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.3&-4.1\\-8.7&3.8\end{matrix}\right))\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3.8}{6.3\times 3.8-\left(-4.1\left(-8.7\right)\right)}&-\frac{-4.1}{6.3\times 3.8-\left(-4.1\left(-8.7\right)\right)}\\-\frac{-8.7}{6.3\times 3.8-\left(-4.1\left(-8.7\right)\right)}&\frac{6.3}{6.3\times 3.8-\left(-4.1\left(-8.7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{380}{1173}&-\frac{410}{1173}\\-\frac{290}{391}&-\frac{210}{391}\end{matrix}\right)\left(\begin{matrix}-3.5\\-8.2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{380}{1173}\left(-3.5\right)-\frac{410}{1173}\left(-8.2\right)\\-\frac{290}{391}\left(-3.5\right)-\frac{210}{391}\left(-8.2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=7$$

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.

$$6.3x-4.1y=-3.5,-8.7x+3.8y=-8.2$$

Kia ōrite ai a \frac{63x}{10} me -\frac{87x}{10}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8.7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.3.

$$-8.7\times 6.3x-8.7\left(-4.1\right)y=-8.7\left(-3.5\right),6.3\left(-8.7\right)x+6.3\times 3.8y=6.3\left(-8.2\right)$$

Whakarūnātia.

$$-54.81x+35.67y=30.45,-54.81x+23.94y=-51.66$$

Me tango -54.81x+23.94y=-51.66 mai i -54.81x+35.67y=30.45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-54.81x+54.81x+35.67y-23.94y=30.45+51.66$$

Tāpiri -\frac{5481x}{100} ki te \frac{5481x}{100}. Ka whakakore atu ngā kupu -\frac{5481x}{100} me \frac{5481x}{100}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$35.67y-23.94y=30.45+51.66$$

Tāpiri \frac{3567y}{100} ki te -\frac{1197y}{50}.

$$11.73y=30.45+51.66$$

Tāpiri 30.45 ki te 51.66 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.

$$11.73y=82.11$$

Whakawehea ngā taha e rua o te whārite ki te 11.73, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki -8.7x+3.8y=-8.2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8.7x+3.8\times 7=-8.2$$

Whakareatia 3.8 ki te 7.

$$-8.7x+26.6=-8.2$$

Me tango 26.6 mai i ngā taha e rua o te whārite.

$$-8.7x=-34.8$$

Whakawehea ngā taha e rua o te whārite ki te -8.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$