Whakaoti i te 3x-5y=4;9x-2y=7

$$3 x - 5 y = 4 ; 9 x - 2 y = 7$$

Whakaoti mō x, y

$$x=\frac{9}{13}\approx 0.692307692$$

$$y=-\frac{5}{13}\approx -0.384615385$$

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.

$$3x-5y=4,9x-2y=7$$

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

$$3x-5y=4$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y+4$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y+4\right)$$

Whakareatia \frac{1}{3} ki te 5y+4.

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

Whakakapia te \frac{5y+4}{3} mō te x ki tērā atu whārite, 9x-2y=7.

$$9\left(\frac{5}{3}y+\frac{4}{3}\right)-2y=7$$

Whakareatia 9 ki te \frac{5y+4}{3}.

$$15y+12-2y=7$$

Tāpiri 15y ki te -2y.

$$13y+12=7$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$13y=-5$$

Whakawehea ngā taha e rua ki te 13.

$$y=-\frac{5}{13}$$

Whakaurua te -\frac{5}{13} mō y ki x=\frac{5}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-\frac{5}{13}\right)+\frac{4}{3}$$

Whakareatia \frac{5}{3} ki te -\frac{5}{13} 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.

$$x=-\frac{25}{39}+\frac{4}{3}$$

Tāpiri \frac{4}{3} ki te -\frac{25}{39} 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.

$$x=\frac{9}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{13},y=-\frac{5}{13}$$

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.

$$3x-5y=4,9x-2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\9&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-5\times 9\right)}&-\frac{-5}{3\left(-2\right)-\left(-5\times 9\right)}\\-\frac{9}{3\left(-2\right)-\left(-5\times 9\right)}&\frac{3}{3\left(-2\right)-\left(-5\times 9\right)}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{39}&\frac{5}{39}\\-\frac{3}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{39}\times 4+\frac{5}{39}\times 7\\-\frac{3}{13}\times 4+\frac{1}{13}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}\\-\frac{5}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{13},y=-\frac{5}{13}$$

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.

$$3x-5y=4,9x-2y=7$$

Kia ōrite ai a 3x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$9\times 3x+9\left(-5\right)y=9\times 4,3\times 9x+3\left(-2\right)y=3\times 7$$

Whakarūnātia.

$$27x-45y=36,27x-6y=21$$

Me tango 27x-6y=21 mai i 27x-45y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27x-27x-45y+6y=36-21$$

Tāpiri 27x ki te -27x. Ka whakakore atu ngā kupu 27x me -27x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-45y+6y=36-21$$

Tāpiri -45y ki te 6y.

$$-39y=36-21$$

Tāpiri 36 ki te -21.

$$-39y=15$$

Whakawehea ngā taha e rua ki te -39.

$$y=-\frac{5}{13}$$

Whakaurua te -\frac{5}{13} mō y ki 9x-2y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-2\left(-\frac{5}{13}\right)=7$$

Whakareatia -2 ki te -\frac{5}{13}.

$$9x+\frac{10}{13}=7$$

Me tango \frac{10}{13} mai i ngā taha e rua o te whārite.

$$9x=\frac{81}{13}$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{9}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{13},y=-\frac{5}{13}$$