Whakaoti i te {l}{3x-5y=-16}{2x+5y=31}

$$\left. \begin{array} { l } { 3 x - 5 y = - 16 } \\ { 2 x + 5 y = 31 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=5$$

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.

$$3x-5y=-16,2x+5y=31$$

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

$$3x-5y=-16$$

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

$$3x=5y-16$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y-16\right)$$

Whakareatia \frac{1}{3} ki te 5y-16.

$$x=\frac{5}{3}y-\frac{16}{3}$$

Whakakapia te \frac{5y-16}{3} mō te x ki tērā atu whārite, 2x+5y=31.

$$2\left(\frac{5}{3}y-\frac{16}{3}\right)+5y=31$$

Whakareatia 2 ki te \frac{5y-16}{3}.

$$\frac{10}{3}y-\frac{32}{3}+5y=31$$

Tāpiri \frac{10y}{3} ki te 5y.

$$\frac{25}{3}y-\frac{32}{3}=31$$

Me tāpiri \frac{32}{3} ki ngā taha e rua o te whārite.

$$\frac{25}{3}y=\frac{125}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{5}{3}y-\frac{16}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\times 5-\frac{16}{3}$$

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

$$x=\frac{25-16}{3}$$

Tāpiri -\frac{16}{3} ki te \frac{25}{3} 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.

$$x=3,y=5$$

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.

$$3x-5y=-16,2x+5y=31$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\31\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-5\\2&5\end{matrix}\right))\left(\begin{matrix}3&-5\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&5\end{matrix}\right))\left(\begin{matrix}-16\\31\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&5\end{matrix}\right))\left(\begin{matrix}-16\\31\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&5\end{matrix}\right))\left(\begin{matrix}-16\\31\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-\left(-5\times 2\right)}&-\frac{-5}{3\times 5-\left(-5\times 2\right)}\\-\frac{2}{3\times 5-\left(-5\times 2\right)}&\frac{3}{3\times 5-\left(-5\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-16\\31\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{2}{25}&\frac{3}{25}\end{matrix}\right)\left(\begin{matrix}-16\\31\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-16\right)+\frac{1}{5}\times 31\\-\frac{2}{25}\left(-16\right)+\frac{3}{25}\times 31\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=5$$

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.

$$3x-5y=-16,2x+5y=31$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-5\right)y=2\left(-16\right),3\times 2x+3\times 5y=3\times 31$$

Whakarūnātia.

$$6x-10y=-32,6x+15y=93$$

Me tango 6x+15y=93 mai i 6x-10y=-32 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-10y-15y=-32-93$$

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.

$$-10y-15y=-32-93$$

Tāpiri -10y ki te -15y.

$$-25y=-32-93$$

Tāpiri -32 ki te -93.

$$-25y=-125$$

Whakawehea ngā taha e rua ki te -25.

$$y=5$$

Whakaurua te 5 mō y ki 2x+5y=31. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times 5=31$$

Whakareatia 5 ki te 5.

$$2x+25=31$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=5$$