Whakaoti i te {r}{5x+4y=1}{x-6y=7}

$$\left.\begin{array} { r } { 5 x + 4 y = 1 } \\ { x - 6 y = 7 } \end{array} \right\}$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$5x+4y=1,x-6y=7$$

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

$$5x+4y=1$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$5x=-4y+1$$

Whakawehea ngā taha e rua ki te 5.

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

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

$$x=-\frac{4}{5}y+\frac{1}{5}$$

Whakakapia te \frac{-4y+1}{5} mō te x ki tērā atu whārite, x-6y=7.

$$-\frac{4}{5}y+\frac{1}{5}-6y=7$$

Tāpiri -\frac{4y}{5} ki te -6y.

$$-\frac{34}{5}y+\frac{1}{5}=7$$

Me tango \frac{1}{5} mai i ngā taha e rua o te whārite.

$$-\frac{34}{5}y=\frac{34}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{34}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{4}{5}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{4+1}{5}$$

Tāpiri \frac{1}{5} ki te \frac{4}{5} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$

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.

$$5x+4y=1,x-6y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&4\\1&-6\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}&\frac{2}{17}\\\frac{1}{34}&-\frac{5}{34}\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{17}+\frac{2}{17}\times 7\\\frac{1}{34}-\frac{5}{34}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

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.

$$5x+4y=1,x-6y=7$$

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.

$$5x+4y=1,5x+5\left(-6\right)y=5\times 7$$

Whakarūnātia.

$$5x+4y=1,5x-30y=35$$

Me tango 5x-30y=35 mai i 5x+4y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+4y+30y=1-35$$

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.

$$4y+30y=1-35$$

Tāpiri 4y ki te 30y.

$$34y=1-35$$

Tāpiri 1 ki te -35.

$$34y=-34$$

Whakawehea ngā taha e rua ki te 34.

$$y=-1$$

Whakaurua te -1 mō y ki x-6y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-6\left(-1\right)=7$$

Whakareatia -6 ki te -1.

$$x+6=7$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$