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

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

Whakaoti mō x, y

$$x = \frac{16}{11} = 1\frac{5}{11} \approx 1.454545455$$

$$y=-\frac{1}{11}\approx -0.090909091$$

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.

$$x+5y=1,3x+4y=4$$

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

$$x+5y=1$$

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

$$x=-5y+1$$

Whakakapia te -5y+1 mō te x ki tērā atu whārite, 3x+4y=4.

$$3\left(-5y+1\right)+4y=4$$

Whakareatia 3 ki te -5y+1.

$$-15y+3+4y=4$$

Tāpiri -15y ki te 4y.

$$-11y+3=4$$

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

$$-11y=1$$

Whakawehea ngā taha e rua ki te -11.

$$y=-\frac{1}{11}$$

Whakaurua te -\frac{1}{11} mō y ki x=-5y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-\frac{1}{11}\right)+1$$

Whakareatia -5 ki te -\frac{1}{11}.

$$x=\frac{5}{11}+1$$

Tāpiri 1 ki te \frac{5}{11}.

$$x=\frac{16}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{11},y=-\frac{1}{11}$$

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.

$$x+5y=1,3x+4y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&5\\3&4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}&\frac{5}{11}\\\frac{3}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}1\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}+\frac{5}{11}\times 4\\\frac{3}{11}-\frac{1}{11}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{11}\\-\frac{1}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{11},y=-\frac{1}{11}$$

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.

$$x+5y=1,3x+4y=4$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times 5y=3,3x+4y=4$$

Whakarūnātia.

$$3x+15y=3,3x+4y=4$$

Me tango 3x+4y=4 mai i 3x+15y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+15y-4y=3-4$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-4y=3-4$$

Tāpiri 15y ki te -4y.

$$11y=3-4$$

Tāpiri 3 ki te -4.

$$11y=-1$$

Whakawehea ngā taha e rua ki te 11.

$$y=-\frac{1}{11}$$

Whakaurua te -\frac{1}{11} mō y ki 3x+4y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\left(-\frac{1}{11}\right)=4$$

Whakareatia 4 ki te -\frac{1}{11}.

$$3x-\frac{4}{11}=4$$

Me tāpiri \frac{4}{11} ki ngā taha e rua o te whārite.

$$3x=\frac{48}{11}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{16}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{11},y=-\frac{1}{11}$$