Whakaoti i te {l}{50x+3y=1}{2x-4y=5}

$$\left. \begin{array} { l } { 50 x + 3 y = 1 } \\ { 2 x - 4 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{19}{206}\approx 0.09223301$$

$$y = -\frac{124}{103} = -1\frac{21}{103} \approx -1.203883495$$

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.

$$50x+3y=1,2x-4y=5$$

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

$$50x+3y=1$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$50x=-3y+1$$

Whakawehea ngā taha e rua ki te 50.

$$x=\frac{1}{50}\left(-3y+1\right)$$

Whakareatia \frac{1}{50} ki te -3y+1.

$$x=-\frac{3}{50}y+\frac{1}{50}$$

Whakakapia te \frac{-3y+1}{50} mō te x ki tērā atu whārite, 2x-4y=5.

$$2\left(-\frac{3}{50}y+\frac{1}{50}\right)-4y=5$$

Whakareatia 2 ki te \frac{-3y+1}{50}.

$$-\frac{3}{25}y+\frac{1}{25}-4y=5$$

Tāpiri -\frac{3y}{25} ki te -4y.

$$-\frac{103}{25}y+\frac{1}{25}=5$$

Me tango \frac{1}{25} mai i ngā taha e rua o te whārite.

$$-\frac{103}{25}y=\frac{124}{25}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{103}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{124}{103}$$

Whakaurua te -\frac{124}{103} mō y ki x=-\frac{3}{50}y+\frac{1}{50}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{50}\left(-\frac{124}{103}\right)+\frac{1}{50}$$

Whakareatia -\frac{3}{50} ki te -\frac{124}{103} 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.

$$x=\frac{186}{2575}+\frac{1}{50}$$

Tāpiri \frac{1}{50} ki te \frac{186}{2575} 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.

$$x=\frac{19}{206}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{206},y=-\frac{124}{103}$$

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.

$$50x+3y=1,2x-4y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}50&3\\2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}50&3\\2&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}50&3\\2&-4\end{matrix}\right))\left(\begin{matrix}50&3\\2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&3\\2&-4\end{matrix}\right))\left(\begin{matrix}1\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&3\\2&-4\end{matrix}\right))\left(\begin{matrix}1\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&3\\2&-4\end{matrix}\right))\left(\begin{matrix}1\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{50\left(-4\right)-3\times 2}&-\frac{3}{50\left(-4\right)-3\times 2}\\-\frac{2}{50\left(-4\right)-3\times 2}&\frac{50}{50\left(-4\right)-3\times 2}\end{matrix}\right)\left(\begin{matrix}1\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{103}&\frac{3}{206}\\\frac{1}{103}&-\frac{25}{103}\end{matrix}\right)\left(\begin{matrix}1\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{103}+\frac{3}{206}\times 5\\\frac{1}{103}-\frac{25}{103}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{206}\\-\frac{124}{103}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19}{206},y=-\frac{124}{103}$$

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.

$$50x+3y=1,2x-4y=5$$

Kia ōrite ai a 50x 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 50.

$$2\times 50x+2\times 3y=2,50\times 2x+50\left(-4\right)y=50\times 5$$

Whakarūnātia.

$$100x+6y=2,100x-200y=250$$

Me tango 100x-200y=250 mai i 100x+6y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$100x-100x+6y+200y=2-250$$

Tāpiri 100x ki te -100x. Ka whakakore atu ngā kupu 100x me -100x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+200y=2-250$$

Tāpiri 6y ki te 200y.

$$206y=2-250$$

Tāpiri 2 ki te -250.

$$206y=-248$$

Whakawehea ngā taha e rua ki te 206.

$$y=-\frac{124}{103}$$

Whakaurua te -\frac{124}{103} mō y ki 2x-4y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-4\left(-\frac{124}{103}\right)=5$$

Whakareatia -4 ki te -\frac{124}{103}.

$$2x+\frac{496}{103}=5$$

Me tango \frac{496}{103} mai i ngā taha e rua o te whārite.

$$2x=\frac{19}{103}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{19}{206}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{206},y=-\frac{124}{103}$$