Whakaoti i te {l}{5x-7y=4}{-x+2y=-3}

$$\left. \begin{array} { l } { 5 x - 7 y = 4 } \\ { - x + 2 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{13}{3} = -4\frac{1}{3} \approx -4.333333333$$

$$y = -\frac{11}{3} = -3\frac{2}{3} \approx -3.666666667$$

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.

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

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

$$5x-7y=4$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$5x=7y+4$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(7y+4\right)$$

Whakareatia \frac{1}{5} ki te 7y+4.

$$x=\frac{7}{5}y+\frac{4}{5}$$

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

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

Whakareatia -1 ki te \frac{7y+4}{5}.

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

Tāpiri -\frac{7y}{5} ki te 2y.

$$\frac{3}{5}y-\frac{4}{5}=-3$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$\frac{3}{5}y=-\frac{11}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{11}{3}$$

Whakaurua te -\frac{11}{3} mō y ki x=\frac{7}{5}y+\frac{4}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{5}\left(-\frac{11}{3}\right)+\frac{4}{5}$$

Whakareatia \frac{7}{5} ki te -\frac{11}{3} 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.

$$x=-\frac{77}{15}+\frac{4}{5}$$

Tāpiri \frac{4}{5} ki te -\frac{77}{15} 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.

$$x=-\frac{13}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{13}{3},y=-\frac{11}{3}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-7\\-1&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{3}\\-\frac{11}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{13}{3},y=-\frac{11}{3}$$

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.

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

Kia ōrite ai a 5x 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 5.

$$-5x-\left(-7y\right)=-4,5\left(-1\right)x+5\times 2y=5\left(-3\right)$$

Whakarūnātia.

$$-5x+7y=-4,-5x+10y=-15$$

Me tango -5x+10y=-15 mai i -5x+7y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x+7y-10y=-4+15$$

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y-10y=-4+15$$

Tāpiri 7y ki te -10y.

$$-3y=-4+15$$

Tāpiri -4 ki te 15.

$$-3y=11$$

Whakawehea ngā taha e rua ki te -3.

$$y=-\frac{11}{3}$$

Whakaurua te -\frac{11}{3} mō y ki -x+2y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+2\left(-\frac{11}{3}\right)=-3$$

Whakareatia 2 ki te -\frac{11}{3}.

$$-x-\frac{22}{3}=-3$$

Me tāpiri \frac{22}{3} ki ngā taha e rua o te whārite.

$$-x=\frac{13}{3}$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\frac{13}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{13}{3},y=-\frac{11}{3}$$