Whakaoti i te {l}{3x+5y=10}{2x-y=5}

$$\left. \begin{array} { l } { 3 x + 5 y = 10 } \\ { 2 x - y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{35}{13} = 2\frac{9}{13} \approx 2.692307692$$

$$y=\frac{5}{13}\approx 0.384615385$$

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.

$$3x+5y=10,2x-y=5$$

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

$$3x+5y=10$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y+10$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y+10\right)$$

Whakareatia \frac{1}{3} ki te -5y+10.

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

Whakakapia te \frac{-5y+10}{3} mō te x ki tērā atu whārite, 2x-y=5.

$$2\left(-\frac{5}{3}y+\frac{10}{3}\right)-y=5$$

Whakareatia 2 ki te \frac{-5y+10}{3}.

$$-\frac{10}{3}y+\frac{20}{3}-y=5$$

Tāpiri -\frac{10y}{3} ki te -y.

$$-\frac{13}{3}y+\frac{20}{3}=5$$

Me tango \frac{20}{3} mai i ngā taha e rua o te whārite.

$$-\frac{13}{3}y=-\frac{5}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5}{13}$$

Whakaurua te \frac{5}{13} mō y ki x=-\frac{5}{3}y+\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\times \frac{5}{13}+\frac{10}{3}$$

Whakareatia -\frac{5}{3} ki te \frac{5}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{25}{39}+\frac{10}{3}$$

Tāpiri \frac{10}{3} ki te -\frac{25}{39} 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.

$$x=\frac{35}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{13},y=\frac{5}{13}$$

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.

$$3x+5y=10,2x-y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\2&-1\end{matrix}\right))\left(\begin{matrix}3&5\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&-1\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&-1\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&-1\end{matrix}\right))\left(\begin{matrix}10\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-5\times 2}&-\frac{5}{3\left(-1\right)-5\times 2}\\-\frac{2}{3\left(-1\right)-5\times 2}&\frac{3}{3\left(-1\right)-5\times 2}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{5}{13}\\\frac{2}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}10\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\times 10+\frac{5}{13}\times 5\\\frac{2}{13}\times 10-\frac{3}{13}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{13}\\\frac{5}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{35}{13},y=\frac{5}{13}$$

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.

$$3x+5y=10,2x-y=5$$

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.

$$2\times 3x+2\times 5y=2\times 10,3\times 2x+3\left(-1\right)y=3\times 5$$

Whakarūnātia.

$$6x+10y=20,6x-3y=15$$

Me tango 6x-3y=15 mai i 6x+10y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+10y+3y=20-15$$

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.

$$10y+3y=20-15$$

Tāpiri 10y ki te 3y.

$$13y=20-15$$

Tāpiri 20 ki te -15.

$$13y=5$$

Whakawehea ngā taha e rua ki te 13.

$$y=\frac{5}{13}$$

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

$$2x-\frac{5}{13}=5$$

Me tāpiri \frac{5}{13} ki ngā taha e rua o te whārite.

$$2x=\frac{70}{13}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{35}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{35}{13},y=\frac{5}{13}$$