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

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

Whakaoti mō x, y

$$x=1$$

$$y=0$$

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.

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

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, -6x+5y=-6.

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

Whakareatia -6 ki te y+1.

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

Tāpiri -6y ki te 5y.

$$-y-6=-6$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-y=0$$

Whakawehea ngā taha e rua ki te -1.

$$y=0$$

Whakaurua te 0 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=0$$

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.

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

Kia ōrite ai a 5x me -6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

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

Whakarūnātia.

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

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

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

Tāpiri -30x ki te 30x. Ka whakakore atu ngā kupu -30x me 30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y-25y=-30+30$$

Tāpiri 30y ki te -25y.

$$5y=-30+30$$

Tāpiri -30 ki te 30.

$$5y=0$$

Whakawehea ngā taha e rua ki te 5.

$$y=0$$

Whakaurua te 0 mō y ki -6x+5y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x=-6$$

Whakawehea ngā taha e rua ki te -6.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=0$$