Whakaoti i te {l}{5x+3y=6}{2x+7y=9}

$$\left. \begin{array} { l } { 5 x + 3 y = 6 } \\ { 2 x + 7 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{15}{29}\approx 0.517241379$$

$$y = \frac{33}{29} = 1\frac{4}{29} \approx 1.137931034$$

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.

$$5x+3y=6,2x+7y=9$$

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

$$5x+3y=6$$

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

$$5x=-3y+6$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+6\right)$$

Whakareatia \frac{1}{5} ki te -3y+6.

$$x=-\frac{3}{5}y+\frac{6}{5}$$

Whakakapia te \frac{-3y+6}{5} mō te x ki tērā atu whārite, 2x+7y=9.

$$2\left(-\frac{3}{5}y+\frac{6}{5}\right)+7y=9$$

Whakareatia 2 ki te \frac{-3y+6}{5}.

$$-\frac{6}{5}y+\frac{12}{5}+7y=9$$

Tāpiri -\frac{6y}{5} ki te 7y.

$$\frac{29}{5}y+\frac{12}{5}=9$$

Me tango \frac{12}{5} mai i ngā taha e rua o te whārite.

$$\frac{29}{5}y=\frac{33}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{33}{29}$$

Whakaurua te \frac{33}{29} mō y ki x=-\frac{3}{5}y+\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\times \frac{33}{29}+\frac{6}{5}$$

Whakareatia -\frac{3}{5} ki te \frac{33}{29} 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.

$$x=-\frac{99}{145}+\frac{6}{5}$$

Tāpiri \frac{6}{5} ki te -\frac{99}{145} 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.

$$x=\frac{15}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{29},y=\frac{33}{29}$$

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.

$$5x+3y=6,2x+7y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&3\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&3\\2&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&3\\2&7\end{matrix}\right))\left(\begin{matrix}5&3\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\2&7\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\2&7\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\2&7\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-3\times 2}&-\frac{3}{5\times 7-3\times 2}\\-\frac{2}{5\times 7-3\times 2}&\frac{5}{5\times 7-3\times 2}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{29}&-\frac{3}{29}\\-\frac{2}{29}&\frac{5}{29}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{29}\times 6-\frac{3}{29}\times 9\\-\frac{2}{29}\times 6+\frac{5}{29}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{29}\\\frac{33}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{29},y=\frac{33}{29}$$

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.

$$5x+3y=6,2x+7y=9$$

Kia ōrite ai a 5x 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 5.

$$2\times 5x+2\times 3y=2\times 6,5\times 2x+5\times 7y=5\times 9$$

Whakarūnātia.

$$10x+6y=12,10x+35y=45$$

Me tango 10x+35y=45 mai i 10x+6y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+6y-35y=12-45$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-35y=12-45$$

Tāpiri 6y ki te -35y.

$$-29y=12-45$$

Tāpiri 12 ki te -45.

$$-29y=-33$$

Whakawehea ngā taha e rua ki te -29.

$$y=\frac{33}{29}$$

Whakaurua te \frac{33}{29} mō y ki 2x+7y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+7\times \frac{33}{29}=9$$

Whakareatia 7 ki te \frac{33}{29}.

$$2x+\frac{231}{29}=9$$

Me tango \frac{231}{29} mai i ngā taha e rua o te whārite.

$$2x=\frac{30}{29}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{15}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{29},y=\frac{33}{29}$$