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

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

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.

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

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

$$2x-y=5$$

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

$$2x=y+5$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=\frac{1}{2}y+\frac{5}{2}$$

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

$$3\left(\frac{1}{2}y+\frac{5}{2}\right)+4y=2$$

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

$$\frac{3}{2}y+\frac{15}{2}+4y=2$$

Tāpiri \frac{3y}{2} ki te 4y.

$$\frac{11}{2}y+\frac{15}{2}=2$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{2}, 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{1}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia \frac{1}{2} ki te -1.

$$x=\frac{-1+5}{2}$$

Tāpiri \frac{5}{2} ki te -\frac{1}{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.

$$x=2,y=-1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\3&4\end{matrix}\right).

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-\left(-3\right)}&-\frac{-1}{2\times 4-\left(-3\right)}\\-\frac{3}{2\times 4-\left(-3\right)}&\frac{2}{2\times 4-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}5\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\times 5+\frac{1}{11}\times 2\\-\frac{3}{11}\times 5+\frac{2}{11}\times 2\end{matrix}\right)$$

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.

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

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.

$$3\times 2x+3\left(-1\right)y=3\times 5,2\times 3x+2\times 4y=2\times 2$$

Whakarūnātia.

$$6x-3y=15,6x+8y=4$$

Me tango 6x+8y=4 mai i 6x-3y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-3y-8y=15-4$$

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.

$$-3y-8y=15-4$$

Tāpiri -3y ki te -8y.

$$-11y=15-4$$

Tāpiri 15 ki te -4.

$$-11y=11$$

Whakawehea ngā taha e rua ki te -11.

$$y=-1$$

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

$$3x+4\left(-1\right)=2$$

Whakareatia 4 ki te -1.

$$3x-4=2$$

Me tāpiri 4 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.

$$x=2,y=-1$$