Whakaoti i te 4x-5y=2text{and}x+10y=41

$$4 x - 5 y = 2 \text { and } x + 10 y = 41$$

Whakaoti mō x, y

$$x=5$$

$$y = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

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.

$$4x-5y=2,x+10y=41$$

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

$$4x-5y=2$$

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

$$4x=5y+2$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te 5y+2.

$$x=\frac{5}{4}y+\frac{1}{2}$$

Whakakapia te \frac{5y}{4}+\frac{1}{2} mō te x ki tērā atu whārite, x+10y=41.

$$\frac{5}{4}y+\frac{1}{2}+10y=41$$

Tāpiri \frac{5y}{4} ki te 10y.

$$\frac{45}{4}y+\frac{1}{2}=41$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$\frac{45}{4}y=\frac{81}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{45}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki x=\frac{5}{4}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4}\times \frac{18}{5}+\frac{1}{2}$$

Whakareatia \frac{5}{4} ki te \frac{18}{5} 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.

$$x=\frac{9+1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{9}{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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=\frac{18}{5}$$

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.

$$4x-5y=2,x+10y=41$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\41\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-5\\1&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}4&-5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{4\times 10-\left(-5\right)}&-\frac{-5}{4\times 10-\left(-5\right)}\\-\frac{1}{4\times 10-\left(-5\right)}&\frac{4}{4\times 10-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}2\\41\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{1}{9}\\-\frac{1}{45}&\frac{4}{45}\end{matrix}\right)\left(\begin{matrix}2\\41\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 2+\frac{1}{9}\times 41\\-\frac{1}{45}\times 2+\frac{4}{45}\times 41\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\\frac{18}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=\frac{18}{5}$$

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.

$$4x-5y=2,x+10y=41$$

Kia ōrite ai a 4x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$4x-5y=2,4x+4\times 10y=4\times 41$$

Whakarūnātia.

$$4x-5y=2,4x+40y=164$$

Me tango 4x+40y=164 mai i 4x-5y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x-5y-40y=2-164$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-40y=2-164$$

Tāpiri -5y ki te -40y.

$$-45y=2-164$$

Tāpiri 2 ki te -164.

$$-45y=-162$$

Whakawehea ngā taha e rua ki te -45.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki x+10y=41. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+10\times \frac{18}{5}=41$$

Whakareatia 10 ki te \frac{18}{5}.

$$x+36=41$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=\frac{18}{5}$$