Whakaoti i te {l}{5x-5y=-5}{-5x+6y=1}

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

Whakaoti mō x, y

$$x=-5$$

$$y=-4$$

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.

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

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

$$5x-5y=-5$$

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

$$5x=5y-5$$

Whakawehea ngā taha e rua ki te 5.

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

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

$$x=y-1$$

Whakakapia te y-1 mō te x ki tērā atu whārite, -5x+6y=1.

$$-5\left(y-1\right)+6y=1$$

Whakareatia -5 ki te y-1.

$$-5y+5+6y=1$$

Tāpiri -5y ki te 6y.

$$y+5=1$$

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

$$y=-4$$

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

$$x=-4-1$$

Tāpiri -1 ki te -4.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-4$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=-4$$

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.

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

Kia ōrite ai a 5x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

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

Whakarūnātia.

$$-25x+25y=25,-25x+30y=5$$

Me tango -25x+30y=5 mai i -25x+25y=25 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-25x+25x+25y-30y=25-5$$

Tāpiri -25x ki te 25x. Ka whakakore atu ngā kupu -25x me 25x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y-30y=25-5$$

Tāpiri 25y ki te -30y.

$$-5y=25-5$$

Tāpiri 25 ki te -5.

$$-5y=20$$

Whakawehea ngā taha e rua ki te -5.

$$y=-4$$

Whakaurua te -4 mō y ki -5x+6y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+6\left(-4\right)=1$$

Whakareatia 6 ki te -4.

$$-5x-24=1$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$-5x=25$$

Whakawehea ngā taha e rua ki te -5.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-4$$