Whakaoti i te {l}{2x-5y=10}{4x+y=120}

$$\left. \begin{array} { l } { 2 x - 5 y = 10 } \\ { 4 x + y = 120 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{305}{11} = 27\frac{8}{11} \approx 27.727272727$$

$$y = \frac{100}{11} = 9\frac{1}{11} \approx 9.090909091$$

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.

$$2x-5y=10,4x+y=120$$

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

$$2x-5y=10$$

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

$$2x=5y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(5y+10\right)$$

Whakareatia \frac{1}{2} ki te 10+5y.

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

Whakakapia te 5+\frac{5y}{2} mō te x ki tērā atu whārite, 4x+y=120.

$$4\left(\frac{5}{2}y+5\right)+y=120$$

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

$$10y+20+y=120$$

Tāpiri 10y ki te y.

$$11y+20=120$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$11y=100$$

Whakawehea ngā taha e rua ki te 11.

$$y=\frac{100}{11}$$

Whakaurua te \frac{100}{11} mō y ki x=\frac{5}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{2}\times \frac{100}{11}+5$$

Whakareatia \frac{5}{2} ki te \frac{100}{11} 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.

$$x=\frac{250}{11}+5$$

Tāpiri 5 ki te \frac{250}{11}.

$$x=\frac{305}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{305}{11},y=\frac{100}{11}$$

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.

$$2x-5y=10,4x+y=120$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-5\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\120\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-5\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-5\\4&1\end{matrix}\right))\left(\begin{matrix}2&-5\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&1\end{matrix}\right))\left(\begin{matrix}10\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&1\end{matrix}\right))\left(\begin{matrix}10\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\4&1\end{matrix}\right))\left(\begin{matrix}10\\120\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-5\times 4\right)}&-\frac{-5}{2-\left(-5\times 4\right)}\\-\frac{4}{2-\left(-5\times 4\right)}&\frac{2}{2-\left(-5\times 4\right)}\end{matrix}\right)\left(\begin{matrix}10\\120\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{22}&\frac{5}{22}\\-\frac{2}{11}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}10\\120\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{22}\times 10+\frac{5}{22}\times 120\\-\frac{2}{11}\times 10+\frac{1}{11}\times 120\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{305}{11}\\\frac{100}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{305}{11},y=\frac{100}{11}$$

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.

$$2x-5y=10,4x+y=120$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\left(-5\right)y=4\times 10,2\times 4x+2y=2\times 120$$

Whakarūnātia.

$$8x-20y=40,8x+2y=240$$

Me tango 8x+2y=240 mai i 8x-20y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-20y-2y=40-240$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20y-2y=40-240$$

Tāpiri -20y ki te -2y.

$$-22y=40-240$$

Tāpiri 40 ki te -240.

$$-22y=-200$$

Whakawehea ngā taha e rua ki te -22.

$$y=\frac{100}{11}$$

Whakaurua te \frac{100}{11} mō y ki 4x+y=120. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+\frac{100}{11}=120$$

Me tango \frac{100}{11} mai i ngā taha e rua o te whārite.

$$4x=\frac{1220}{11}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{305}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{305}{11},y=\frac{100}{11}$$