Whakaoti i te {c}{4x+5y=6}{x+7y=3}

$$\left. \begin{array} { c } { 4 x + 5 y = 6 } \\ { x + 7 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{27}{23} = 1\frac{4}{23} \approx 1.173913043$$

$$y=\frac{6}{23}\approx 0.260869565$$

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.

$$4x+5y=6,x+7y=3$$

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

$$4x+5y=6$$

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

$$4x=-5y+6$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+6\right)$$

Whakareatia \frac{1}{4} ki te -5y+6.

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

Whakakapia te -\frac{5y}{4}+\frac{3}{2} mō te x ki tērā atu whārite, x+7y=3.

$$-\frac{5}{4}y+\frac{3}{2}+7y=3$$

Tāpiri -\frac{5y}{4} ki te 7y.

$$\frac{23}{4}y+\frac{3}{2}=3$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$\frac{23}{4}y=\frac{3}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{6}{23}$$

Whakaurua te \frac{6}{23} mō y ki x=-\frac{5}{4}y+\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\times \frac{6}{23}+\frac{3}{2}$$

Whakareatia -\frac{5}{4} ki te \frac{6}{23} 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.

$$x=-\frac{15}{46}+\frac{3}{2}$$

Tāpiri \frac{3}{2} ki te -\frac{15}{46} 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.

$$x=\frac{27}{23}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{23},y=\frac{6}{23}$$

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.

$$4x+5y=6,x+7y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\1&7\end{matrix}\right))\left(\begin{matrix}4&5\\1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\1&7\end{matrix}\right))\left(\begin{matrix}6\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\1&7\end{matrix}\right))\left(\begin{matrix}6\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\1&7\end{matrix}\right))\left(\begin{matrix}6\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4\times 7-5}&-\frac{5}{4\times 7-5}\\-\frac{1}{4\times 7-5}&\frac{4}{4\times 7-5}\end{matrix}\right)\left(\begin{matrix}6\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{23}&-\frac{5}{23}\\-\frac{1}{23}&\frac{4}{23}\end{matrix}\right)\left(\begin{matrix}6\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{23}\times 6-\frac{5}{23}\times 3\\-\frac{1}{23}\times 6+\frac{4}{23}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{27}{23}\\\frac{6}{23}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{27}{23},y=\frac{6}{23}$$

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.

$$4x+5y=6,x+7y=3$$

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.

$$4x+5y=6,4x+4\times 7y=4\times 3$$

Whakarūnātia.

$$4x+5y=6,4x+28y=12$$

Me tango 4x+28y=12 mai i 4x+5y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+5y-28y=6-12$$

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.

$$5y-28y=6-12$$

Tāpiri 5y ki te -28y.

$$-23y=6-12$$

Tāpiri 6 ki te -12.

$$-23y=-6$$

Whakawehea ngā taha e rua ki te -23.

$$y=\frac{6}{23}$$

Whakaurua te \frac{6}{23} mō y ki x+7y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+7\times \frac{6}{23}=3$$

Whakareatia 7 ki te \frac{6}{23}.

$$x+\frac{42}{23}=3$$

Me tango \frac{42}{23} mai i ngā taha e rua o te whārite.

$$x=\frac{27}{23}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{23},y=\frac{6}{23}$$