Whakaoti i te {l}{3x+4y=1}{4x+y=2}

$$\left. \begin{array} { l } { 3 x + 4 y = 1 } \\ { 4 x + y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{7}{13}\approx 0.538461538$$

$$y=-\frac{2}{13}\approx -0.153846154$$

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.

$$3x+4y=1,4x+y=2$$

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

$$3x+4y=1$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$3x=-4y+1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+1\right)$$

Whakareatia \frac{1}{3} ki te -4y+1.

$$x=-\frac{4}{3}y+\frac{1}{3}$$

Whakakapia te \frac{-4y+1}{3} mō te x ki tērā atu whārite, 4x+y=2.

$$4\left(-\frac{4}{3}y+\frac{1}{3}\right)+y=2$$

Whakareatia 4 ki te \frac{-4y+1}{3}.

$$-\frac{16}{3}y+\frac{4}{3}+y=2$$

Tāpiri -\frac{16y}{3} ki te y.

$$-\frac{13}{3}y+\frac{4}{3}=2$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$-\frac{13}{3}y=\frac{2}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{2}{13}$$

Whakaurua te -\frac{2}{13} mō y ki x=-\frac{4}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-\frac{2}{13}\right)+\frac{1}{3}$$

Whakareatia -\frac{4}{3} ki te -\frac{2}{13} 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.

$$x=\frac{8}{39}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{8}{39} 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.

$$x=\frac{7}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{13},y=-\frac{2}{13}$$

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.

$$3x+4y=1,4x+y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\4&1\end{matrix}\right))\left(\begin{matrix}3&4\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&1\end{matrix}\right))\left(\begin{matrix}1\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&1\end{matrix}\right))\left(\begin{matrix}1\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&1\end{matrix}\right))\left(\begin{matrix}1\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-4\times 4}&-\frac{4}{3-4\times 4}\\-\frac{4}{3-4\times 4}&\frac{3}{3-4\times 4}\end{matrix}\right)\left(\begin{matrix}1\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&\frac{4}{13}\\\frac{4}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}1\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}+\frac{4}{13}\times 2\\\frac{4}{13}-\frac{3}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{13}\\-\frac{2}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{7}{13},y=-\frac{2}{13}$$

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.

$$3x+4y=1,4x+y=2$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\times 4y=4,3\times 4x+3y=3\times 2$$

Whakarūnātia.

$$12x+16y=4,12x+3y=6$$

Me tango 12x+3y=6 mai i 12x+16y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+16y-3y=4-6$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16y-3y=4-6$$

Tāpiri 16y ki te -3y.

$$13y=4-6$$

Tāpiri 4 ki te -6.

$$13y=-2$$

Whakawehea ngā taha e rua ki te 13.

$$y=-\frac{2}{13}$$

Whakaurua te -\frac{2}{13} mō y ki 4x+y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-\frac{2}{13}=2$$

Me tāpiri \frac{2}{13} ki ngā taha e rua o te whārite.

$$4x=\frac{28}{13}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{7}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{13},y=-\frac{2}{13}$$