Whakaoti i te {l}{4x+3y=11}{3x-7y=-1}

$$\left\{ \begin{array} { l } { 4 x + 3 y = 11 } \\ { 3 x - 7 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$4x+3y=11,3x-7y=-1$$

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

$$4x+3y=11$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$4x=-3y+11$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+11\right)$$

Whakareatia \frac{1}{4} ki te -3y+11.

$$x=-\frac{3}{4}y+\frac{11}{4}$$

Whakakapia te \frac{-3y+11}{4} mō te x ki tērā atu whārite, 3x-7y=-1.

$$3\left(-\frac{3}{4}y+\frac{11}{4}\right)-7y=-1$$

Whakareatia 3 ki te \frac{-3y+11}{4}.

$$-\frac{9}{4}y+\frac{33}{4}-7y=-1$$

Tāpiri -\frac{9y}{4} ki te -7y.

$$-\frac{37}{4}y+\frac{33}{4}=-1$$

Me tango \frac{33}{4} mai i ngā taha e rua o te whārite.

$$-\frac{37}{4}y=-\frac{37}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{37}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{3}{4}y+\frac{11}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-3+11}{4}$$

Tāpiri \frac{11}{4} ki te -\frac{3}{4} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

$$4x+3y=11,3x-7y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\3&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\3&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\3&-7\end{matrix}\right))\left(\begin{matrix}4&3\\3&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&-7\end{matrix}\right))\left(\begin{matrix}11\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&-7\end{matrix}\right))\left(\begin{matrix}11\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&-7\end{matrix}\right))\left(\begin{matrix}11\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4\left(-7\right)-3\times 3}&-\frac{3}{4\left(-7\right)-3\times 3}\\-\frac{3}{4\left(-7\right)-3\times 3}&\frac{4}{4\left(-7\right)-3\times 3}\end{matrix}\right)\left(\begin{matrix}11\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{37}&\frac{3}{37}\\\frac{3}{37}&-\frac{4}{37}\end{matrix}\right)\left(\begin{matrix}11\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{37}\times 11+\frac{3}{37}\left(-1\right)\\\frac{3}{37}\times 11-\frac{4}{37}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$4x+3y=11,3x-7y=-1$$

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

$$3\times 4x+3\times 3y=3\times 11,4\times 3x+4\left(-7\right)y=4\left(-1\right)$$

Whakarūnātia.

$$12x+9y=33,12x-28y=-4$$

Me tango 12x-28y=-4 mai i 12x+9y=33 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+9y+28y=33+4$$

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.

$$9y+28y=33+4$$

Tāpiri 9y ki te 28y.

$$37y=33+4$$

Tāpiri 33 ki te 4.

$$37y=37$$

Whakawehea ngā taha e rua ki te 37.

$$y=1$$

Whakaurua te 1 mō y ki 3x-7y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-7=-1$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$