Whakaoti i te {l}{5x-y=8}{10x+3y=6}

$$\left. \begin{array} { l } { 5 x - y = 8 } \\ { 10 x + 3 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

$$y=-2$$

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.

$$5x-y=8,10x+3y=6$$

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

$$5x-y=8$$

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

$$5x=y+8$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+8\right)$$

Whakareatia \frac{1}{5} ki te y+8.

$$x=\frac{1}{5}y+\frac{8}{5}$$

Whakakapia te \frac{8+y}{5} mō te x ki tērā atu whārite, 10x+3y=6.

$$10\left(\frac{1}{5}y+\frac{8}{5}\right)+3y=6$$

Whakareatia 10 ki te \frac{8+y}{5}.

$$2y+16+3y=6$$

Tāpiri 2y ki te 3y.

$$5y+16=6$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$5y=-10$$

Whakawehea ngā taha e rua ki te 5.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{1}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\left(-2\right)+\frac{8}{5}$$

Whakareatia \frac{1}{5} ki te -2.

$$x=\frac{-2+8}{5}$$

Tāpiri \frac{8}{5} ki te -\frac{2}{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.

$$x=\frac{6}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=-2$$

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.

$$5x-y=8,10x+3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\10&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\10&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\10&3\end{matrix}\right))\left(\begin{matrix}5&-1\\10&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\10&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\10&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\10&3\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-\left(-10\right)}&-\frac{-1}{5\times 3-\left(-10\right)}\\-\frac{10}{5\times 3-\left(-10\right)}&\frac{5}{5\times 3-\left(-10\right)}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}&\frac{1}{25}\\-\frac{2}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{25}\times 8+\frac{1}{25}\times 6\\-\frac{2}{5}\times 8+\frac{1}{5}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{6}{5},y=-2$$

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.

$$5x-y=8,10x+3y=6$$

Kia ōrite ai a 5x me 10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$10\times 5x+10\left(-1\right)y=10\times 8,5\times 10x+5\times 3y=5\times 6$$

Whakarūnātia.

$$50x-10y=80,50x+15y=30$$

Me tango 50x+15y=30 mai i 50x-10y=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x-10y-15y=80-30$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y-15y=80-30$$

Tāpiri -10y ki te -15y.

$$-25y=80-30$$

Tāpiri 80 ki te -30.

$$-25y=50$$

Whakawehea ngā taha e rua ki te -25.

$$y=-2$$

Whakaurua te -2 mō y ki 10x+3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$10x+3\left(-2\right)=6$$

Whakareatia 3 ki te -2.

$$10x-6=6$$

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

$$10x=12$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{6}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{5},y=-2$$