Whakaoti i te {l}{3x+4y=-4}{4x+3y=6}

$$\left. \begin{array} { l } { 3 x + 4 y = - 4 } \\ { 4 x + 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{36}{7} = 5\frac{1}{7} \approx 5.142857143$$

$$y = -\frac{34}{7} = -4\frac{6}{7} \approx -4.857142857$$

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.

$$3x+4y=-4,4x+3y=6$$

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

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

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

$$3x=-4y-4$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{-4y-4}{3} mō te x ki tērā atu whārite, 4x+3y=6.

$$4\left(-\frac{4}{3}y-\frac{4}{3}\right)+3y=6$$

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

$$-\frac{16}{3}y-\frac{16}{3}+3y=6$$

Tāpiri -\frac{16y}{3} ki te 3y.

$$-\frac{7}{3}y-\frac{16}{3}=6$$

Me tāpiri \frac{16}{3} ki ngā taha e rua o te whārite.

$$-\frac{7}{3}y=\frac{34}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{34}{7}$$

Whakaurua te -\frac{34}{7} mō y ki x=-\frac{4}{3}y-\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-\frac{34}{7}\right)-\frac{4}{3}$$

Whakareatia -\frac{4}{3} ki te -\frac{34}{7} 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.

$$x=\frac{136}{21}-\frac{4}{3}$$

Tāpiri -\frac{4}{3} ki te \frac{136}{21} 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.

$$x=\frac{36}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{36}{7},y=-\frac{34}{7}$$

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.

$$3x+4y=-4,4x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\4&3\end{matrix}\right))\left(\begin{matrix}3&4\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&3\end{matrix}\right))\left(\begin{matrix}-4\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&3\end{matrix}\right))\left(\begin{matrix}-4\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\4&3\end{matrix}\right))\left(\begin{matrix}-4\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-4\times 4}&-\frac{4}{3\times 3-4\times 4}\\-\frac{4}{3\times 3-4\times 4}&\frac{3}{3\times 3-4\times 4}\end{matrix}\right)\left(\begin{matrix}-4\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{7}&\frac{4}{7}\\\frac{4}{7}&-\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}-4\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{7}\left(-4\right)+\frac{4}{7}\times 6\\\frac{4}{7}\left(-4\right)-\frac{3}{7}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{36}{7}\\-\frac{34}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{36}{7},y=-\frac{34}{7}$$

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.

$$3x+4y=-4,4x+3y=6$$

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.

$$4\times 3x+4\times 4y=4\left(-4\right),3\times 4x+3\times 3y=3\times 6$$

Whakarūnātia.

$$12x+16y=-16,12x+9y=18$$

Me tango 12x+9y=18 mai i 12x+16y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+16y-9y=-16-18$$

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.

$$16y-9y=-16-18$$

Tāpiri 16y ki te -9y.

$$7y=-16-18$$

Tāpiri -16 ki te -18.

$$7y=-34$$

Whakawehea ngā taha e rua ki te 7.

$$y=-\frac{34}{7}$$

Whakaurua te -\frac{34}{7} mō y ki 4x+3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\left(-\frac{34}{7}\right)=6$$

Whakareatia 3 ki te -\frac{34}{7}.

$$4x-\frac{102}{7}=6$$

Me tāpiri \frac{102}{7} ki ngā taha e rua o te whārite.

$$4x=\frac{144}{7}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{36}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{36}{7},y=-\frac{34}{7}$$