Whakaoti i te {l}{4x+3y=18}{x+5y=2}

$$\left. \begin{array} { l } { 4 x + 3 y = 18 } \\ { x + 5 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{84}{17} = 4\frac{16}{17} \approx 4.941176471$$

$$y=-\frac{10}{17}\approx -0.588235294$$

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.

$$4x+3y=18,x+5y=2$$

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

$$4x+3y=18$$

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

$$4x=-3y+18$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+18\right)$$

Whakareatia \frac{1}{4} ki te -3y+18.

$$x=-\frac{3}{4}y+\frac{9}{2}$$

Whakakapia te -\frac{3y}{4}+\frac{9}{2} mō te x ki tērā atu whārite, x+5y=2.

$$-\frac{3}{4}y+\frac{9}{2}+5y=2$$

Tāpiri -\frac{3y}{4} ki te 5y.

$$\frac{17}{4}y+\frac{9}{2}=2$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$\frac{17}{4}y=-\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{10}{17}$$

Whakaurua te -\frac{10}{17} mō y ki x=-\frac{3}{4}y+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-\frac{10}{17}\right)+\frac{9}{2}$$

Whakareatia -\frac{3}{4} ki te -\frac{10}{17} 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.

$$x=\frac{15}{34}+\frac{9}{2}$$

Tāpiri \frac{9}{2} ki te \frac{15}{34} 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.

$$x=\frac{84}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{84}{17},y=-\frac{10}{17}$$

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.

$$4x+3y=18,x+5y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&3\\1&5\end{matrix}\right))\left(\begin{matrix}4&3\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-3}&-\frac{3}{4\times 5-3}\\-\frac{1}{4\times 5-3}&\frac{4}{4\times 5-3}\end{matrix}\right)\left(\begin{matrix}18\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}&-\frac{3}{17}\\-\frac{1}{17}&\frac{4}{17}\end{matrix}\right)\left(\begin{matrix}18\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}\times 18-\frac{3}{17}\times 2\\-\frac{1}{17}\times 18+\frac{4}{17}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{84}{17}\\-\frac{10}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{84}{17},y=-\frac{10}{17}$$

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.

$$4x+3y=18,x+5y=2$$

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.

$$4x+3y=18,4x+4\times 5y=4\times 2$$

Whakarūnātia.

$$4x+3y=18,4x+20y=8$$

Me tango 4x+20y=8 mai i 4x+3y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+3y-20y=18-8$$

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.

$$3y-20y=18-8$$

Tāpiri 3y ki te -20y.

$$-17y=18-8$$

Tāpiri 18 ki te -8.

$$-17y=10$$

Whakawehea ngā taha e rua ki te -17.

$$y=-\frac{10}{17}$$

Whakaurua te -\frac{10}{17} mō y ki x+5y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5\left(-\frac{10}{17}\right)=2$$

Whakareatia 5 ki te -\frac{10}{17}.

$$x-\frac{50}{17}=2$$

Me tāpiri \frac{50}{17} ki ngā taha e rua o te whārite.

$$x=\frac{84}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{84}{17},y=-\frac{10}{17}$$