Whakaoti i te 5x-2y=1;3x+5y=13

$$5 x - 2 y = 1 ; 3 x + 5 y = 13$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$5x-2y=1,3x+5y=13$$

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

$$5x-2y=1$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$5x=2y+1$$

Whakawehea ngā taha e rua ki te 5.

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

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

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

Whakakapia te \frac{2y+1}{5} mō te x ki tērā atu whārite, 3x+5y=13.

$$3\left(\frac{2}{5}y+\frac{1}{5}\right)+5y=13$$

Whakareatia 3 ki te \frac{2y+1}{5}.

$$\frac{6}{5}y+\frac{3}{5}+5y=13$$

Tāpiri \frac{6y}{5} ki te 5y.

$$\frac{31}{5}y+\frac{3}{5}=13$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$\frac{31}{5}y=\frac{62}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{31}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{2}{5}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{4+1}{5}$$

Tāpiri \frac{1}{5} ki te \frac{4}{5} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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.

$$5x-2y=1,3x+5y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\13\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}5&-2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-\left(-2\times 3\right)}&-\frac{-2}{5\times 5-\left(-2\times 3\right)}\\-\frac{3}{5\times 5-\left(-2\times 3\right)}&\frac{5}{5\times 5-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}1\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}&\frac{2}{31}\\-\frac{3}{31}&\frac{5}{31}\end{matrix}\right)\left(\begin{matrix}1\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}+\frac{2}{31}\times 13\\-\frac{3}{31}+\frac{5}{31}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$5x-2y=1,3x+5y=13$$

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3\left(-2\right)y=3,5\times 3x+5\times 5y=5\times 13$$

Whakarūnātia.

$$15x-6y=3,15x+25y=65$$

Me tango 15x+25y=65 mai i 15x-6y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-6y-25y=3-65$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-25y=3-65$$

Tāpiri -6y ki te -25y.

$$-31y=3-65$$

Tāpiri 3 ki te -65.

$$-31y=-62$$

Whakawehea ngā taha e rua ki te -31.

$$y=2$$

Whakaurua te 2 mō y ki 3x+5y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\times 2=13$$

Whakareatia 5 ki te 2.

$$3x+10=13$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$