Whakaoti i te {l}{2x-5y=-11}{3x+4y=18}

$$\left\{ \begin{array} { l } { 2 x - 5 y = - 11 } \\ { 3 x + 4 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$2x-5y=-11,3x+4y=18$$

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

$$2x-5y=-11$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$2x=5y-11$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(5y-11\right)$$

Whakareatia \frac{1}{2} ki te 5y-11.

$$x=\frac{5}{2}y-\frac{11}{2}$$

Whakakapia te \frac{5y-11}{2} mō te x ki tērā atu whārite, 3x+4y=18.

$$3\left(\frac{5}{2}y-\frac{11}{2}\right)+4y=18$$

Whakareatia 3 ki te \frac{5y-11}{2}.

$$\frac{15}{2}y-\frac{33}{2}+4y=18$$

Tāpiri \frac{15y}{2} ki te 4y.

$$\frac{23}{2}y-\frac{33}{2}=18$$

Me tāpiri \frac{33}{2} ki ngā taha e rua o te whārite.

$$\frac{23}{2}y=\frac{69}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{5}{2}y-\frac{11}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{2}\times 3-\frac{11}{2}$$

Whakareatia \frac{5}{2} ki te 3.

$$x=\frac{15-11}{2}$$

Tāpiri -\frac{11}{2} ki te \frac{15}{2} 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.

$$x=2,y=3$$

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.

$$2x-5y=-11,3x+4y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-5\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-5\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-5\\3&4\end{matrix}\right))\left(\begin{matrix}2&-5\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\3&4\end{matrix}\right))\left(\begin{matrix}-11\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\3&4\end{matrix}\right))\left(\begin{matrix}-11\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\3&4\end{matrix}\right))\left(\begin{matrix}-11\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-\left(-5\times 3\right)}&-\frac{-5}{2\times 4-\left(-5\times 3\right)}\\-\frac{3}{2\times 4-\left(-5\times 3\right)}&\frac{2}{2\times 4-\left(-5\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-11\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{23}&\frac{5}{23}\\-\frac{3}{23}&\frac{2}{23}\end{matrix}\right)\left(\begin{matrix}-11\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{23}\left(-11\right)+\frac{5}{23}\times 18\\-\frac{3}{23}\left(-11\right)+\frac{2}{23}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$2x-5y=-11,3x+4y=18$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\left(-5\right)y=3\left(-11\right),2\times 3x+2\times 4y=2\times 18$$

Whakarūnātia.

$$6x-15y=-33,6x+8y=36$$

Me tango 6x+8y=36 mai i 6x-15y=-33 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-15y-8y=-33-36$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-8y=-33-36$$

Tāpiri -15y ki te -8y.

$$-23y=-33-36$$

Tāpiri -33 ki te -36.

$$-23y=-69$$

Whakawehea ngā taha e rua ki te -23.

$$y=3$$

Whakaurua te 3 mō y ki 3x+4y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\times 3=18$$

Whakareatia 4 ki te 3.

$$3x+12=18$$

Me tango 12 mai i 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.

$$x=2,y=3$$