Whakaoti i te {l}{4x+9y=-21}{3x+4y=-13}

$$\left. \begin{array} { l } { 4 x + 9 y = - 21 } \\ { 3 x + 4 y = - 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-1$$

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.

$$4x+9y=-21,3x+4y=-13$$

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

$$4x+9y=-21$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$4x=-9y-21$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-9y-21\right)$$

Whakareatia \frac{1}{4} ki te -9y-21.

$$x=-\frac{9}{4}y-\frac{21}{4}$$

Whakakapia te \frac{-9y-21}{4} mō te x ki tērā atu whārite, 3x+4y=-13.

$$3\left(-\frac{9}{4}y-\frac{21}{4}\right)+4y=-13$$

Whakareatia 3 ki te \frac{-9y-21}{4}.

$$-\frac{27}{4}y-\frac{63}{4}+4y=-13$$

Tāpiri -\frac{27y}{4} ki te 4y.

$$-\frac{11}{4}y-\frac{63}{4}=-13$$

Me tāpiri \frac{63}{4} ki ngā taha e rua o te whārite.

$$-\frac{11}{4}y=\frac{11}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{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{9}{4}y-\frac{21}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{4}\left(-1\right)-\frac{21}{4}$$

Whakareatia -\frac{9}{4} ki te -1.

$$x=\frac{9-21}{4}$$

Tāpiri -\frac{21}{4} ki te \frac{9}{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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-1$$

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.

$$4x+9y=-21,3x+4y=-13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&9\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&9\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&9\\3&4\end{matrix}\right))\left(\begin{matrix}4&9\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&4\end{matrix}\right))\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&4\end{matrix}\right))\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\3&4\end{matrix}\right))\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4\times 4-9\times 3}&-\frac{9}{4\times 4-9\times 3}\\-\frac{3}{4\times 4-9\times 3}&\frac{4}{4\times 4-9\times 3}\end{matrix}\right)\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}&\frac{9}{11}\\\frac{3}{11}&-\frac{4}{11}\end{matrix}\right)\left(\begin{matrix}-21\\-13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}\left(-21\right)+\frac{9}{11}\left(-13\right)\\\frac{3}{11}\left(-21\right)-\frac{4}{11}\left(-13\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-1$$

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.

$$4x+9y=-21,3x+4y=-13$$

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.

$$3\times 4x+3\times 9y=3\left(-21\right),4\times 3x+4\times 4y=4\left(-13\right)$$

Whakarūnātia.

$$12x+27y=-63,12x+16y=-52$$

Me tango 12x+16y=-52 mai i 12x+27y=-63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+27y-16y=-63+52$$

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.

$$27y-16y=-63+52$$

Tāpiri 27y ki te -16y.

$$11y=-63+52$$

Tāpiri -63 ki te 52.

$$11y=-11$$

Whakawehea ngā taha e rua ki te 11.

$$y=-1$$

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

$$3x+4\left(-1\right)=-13$$

Whakareatia 4 ki te -1.

$$3x-4=-13$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$3x=-9$$

Whakawehea ngā taha e rua ki te 3.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-1$$