Whakaoti i te {l}{13=8x-y}{8=x-10y}

$$\left. \begin{array} { l } { 13 = 8 x - y } \\ { 8 = x - 10 y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{122}{79} = 1\frac{43}{79} \approx 1.544303797$$

$$y=-\frac{51}{79}\approx -0.64556962$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8x-y=13$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x-10y=8$$

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.

$$8x-y=13,x-10y=8$$

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

$$8x-y=13$$

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

$$8x=y+13$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(y+13\right)$$

Whakareatia \frac{1}{8} ki te y+13.

$$x=\frac{1}{8}y+\frac{13}{8}$$

Whakakapia te \frac{13+y}{8} mō te x ki tērā atu whārite, x-10y=8.

$$\frac{1}{8}y+\frac{13}{8}-10y=8$$

Tāpiri \frac{y}{8} ki te -10y.

$$-\frac{79}{8}y+\frac{13}{8}=8$$

Me tango \frac{13}{8} mai i ngā taha e rua o te whārite.

$$-\frac{79}{8}y=\frac{51}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{79}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{51}{79}$$

Whakaurua te -\frac{51}{79} mō y ki x=\frac{1}{8}y+\frac{13}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{8}\left(-\frac{51}{79}\right)+\frac{13}{8}$$

Whakareatia \frac{1}{8} ki te -\frac{51}{79} 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.

$$x=-\frac{51}{632}+\frac{13}{8}$$

Tāpiri \frac{13}{8} ki te -\frac{51}{632} 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.

$$x=\frac{122}{79}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{122}{79},y=-\frac{51}{79}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8x-y=13$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x-10y=8$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$8x-y=13,x-10y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-1\\1&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right))\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right))\left(\begin{matrix}13\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right))\left(\begin{matrix}13\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-1\\1&-10\end{matrix}\right))\left(\begin{matrix}13\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{8\left(-10\right)-\left(-1\right)}&-\frac{-1}{8\left(-10\right)-\left(-1\right)}\\-\frac{1}{8\left(-10\right)-\left(-1\right)}&\frac{8}{8\left(-10\right)-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}13\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{79}&-\frac{1}{79}\\\frac{1}{79}&-\frac{8}{79}\end{matrix}\right)\left(\begin{matrix}13\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{79}\times 13-\frac{1}{79}\times 8\\\frac{1}{79}\times 13-\frac{8}{79}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{122}{79}\\-\frac{51}{79}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{122}{79},y=-\frac{51}{79}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$8x-y=13$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x-10y=8$$

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.

$$8x-y=13,x-10y=8$$

Kia ōrite ai a 8x 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 8.

$$8x-y=13,8x+8\left(-10\right)y=8\times 8$$

Whakarūnātia.

$$8x-y=13,8x-80y=64$$

Me tango 8x-80y=64 mai i 8x-y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-y+80y=13-64$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-y+80y=13-64$$

Tāpiri -y ki te 80y.

$$79y=13-64$$

Tāpiri 13 ki te -64.

$$79y=-51$$

Whakawehea ngā taha e rua ki te 79.

$$y=-\frac{51}{79}$$

Whakaurua te -\frac{51}{79} mō y ki x-10y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-10\left(-\frac{51}{79}\right)=8$$

Whakareatia -10 ki te -\frac{51}{79}.

$$x+\frac{510}{79}=8$$

Me tango \frac{510}{79} mai i ngā taha e rua o te whārite.

$$x=\frac{122}{79}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{122}{79},y=-\frac{51}{79}$$