Whakaoti i te {l}{7x+3y=4}{2x+4y=8}

$$\left. \begin{array} { l } { 7 x + 3 y = 4 } \\ { 2 x + 4 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{4}{11}\approx -0.363636364$$

$$y = \frac{24}{11} = 2\frac{2}{11} \approx 2.181818182$$

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.

$$7x+3y=4,2x+4y=8$$

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

$$7x+3y=4$$

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

$$7x=-3y+4$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-3y+4\right)$$

Whakareatia \frac{1}{7} ki te -3y+4.

$$x=-\frac{3}{7}y+\frac{4}{7}$$

Whakakapia te \frac{-3y+4}{7} mō te x ki tērā atu whārite, 2x+4y=8.

$$2\left(-\frac{3}{7}y+\frac{4}{7}\right)+4y=8$$

Whakareatia 2 ki te \frac{-3y+4}{7}.

$$-\frac{6}{7}y+\frac{8}{7}+4y=8$$

Tāpiri -\frac{6y}{7} ki te 4y.

$$\frac{22}{7}y+\frac{8}{7}=8$$

Me tango \frac{8}{7} mai i ngā taha e rua o te whārite.

$$\frac{22}{7}y=\frac{48}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{22}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{24}{11}$$

Whakaurua te \frac{24}{11} mō y ki x=-\frac{3}{7}y+\frac{4}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{7}\times \frac{24}{11}+\frac{4}{7}$$

Whakareatia -\frac{3}{7} ki te \frac{24}{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.

$$x=-\frac{72}{77}+\frac{4}{7}$$

Tāpiri \frac{4}{7} ki te -\frac{72}{77} 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.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{11},y=\frac{24}{11}$$

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.

$$7x+3y=4,2x+4y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&3\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&3\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&3\\2&4\end{matrix}\right))\left(\begin{matrix}7&3\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\2&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\2&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\2&4\end{matrix}\right))\left(\begin{matrix}4\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7\times 4-3\times 2}&-\frac{3}{7\times 4-3\times 2}\\-\frac{2}{7\times 4-3\times 2}&\frac{7}{7\times 4-3\times 2}\end{matrix}\right)\left(\begin{matrix}4\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&-\frac{3}{22}\\-\frac{1}{11}&\frac{7}{22}\end{matrix}\right)\left(\begin{matrix}4\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 4-\frac{3}{22}\times 8\\-\frac{1}{11}\times 4+\frac{7}{22}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{11}\\\frac{24}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{4}{11},y=\frac{24}{11}$$

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.

$$7x+3y=4,2x+4y=8$$

Kia ōrite ai a 7x 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 7.

$$2\times 7x+2\times 3y=2\times 4,7\times 2x+7\times 4y=7\times 8$$

Whakarūnātia.

$$14x+6y=8,14x+28y=56$$

Me tango 14x+28y=56 mai i 14x+6y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+6y-28y=8-56$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-28y=8-56$$

Tāpiri 6y ki te -28y.

$$-22y=8-56$$

Tāpiri 8 ki te -56.

$$-22y=-48$$

Whakawehea ngā taha e rua ki te -22.

$$y=\frac{24}{11}$$

Whakaurua te \frac{24}{11} mō y ki 2x+4y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times \frac{24}{11}=8$$

Whakareatia 4 ki te \frac{24}{11}.

$$2x+\frac{96}{11}=8$$

Me tango \frac{96}{11} mai i ngā taha e rua o te whārite.

$$2x=-\frac{8}{11}$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{11},y=\frac{24}{11}$$