Whakaoti i te {l}{3u+5x=8}{5u+5x=14}

$$\left. \begin{array} { l } { 3 u + 5 x = 8 } \\ { 5 u + 5 x = 14 } \end{array} \right.$$

Whakaoti mō u, x

$$x=-\frac{1}{5}=-0.2$$

$$u=3$$

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.

$$3u+5x=8,5u+5x=14$$

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

$$3u+5x=8$$

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

$$3u=-5x+8$$

Whakawehea ngā taha e rua ki te 3.

$$u=\frac{1}{3}\left(-5x+8\right)$$

Whakareatia \frac{1}{3} ki te -5x+8.

$$u=-\frac{5}{3}x+\frac{8}{3}$$

Whakakapia te \frac{-5x+8}{3} mō te u ki tērā atu whārite, 5u+5x=14.

$$5\left(-\frac{5}{3}x+\frac{8}{3}\right)+5x=14$$

Whakareatia 5 ki te \frac{-5x+8}{3}.

$$-\frac{25}{3}x+\frac{40}{3}+5x=14$$

Tāpiri -\frac{25x}{3} ki te 5x.

$$-\frac{10}{3}x+\frac{40}{3}=14$$

Me tango \frac{40}{3} mai i ngā taha e rua o te whārite.

$$-\frac{10}{3}x=\frac{2}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{10}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{1}{5}$$

Whakaurua te -\frac{1}{5} mō x ki u=-\frac{5}{3}x+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$u=-\frac{5}{3}\left(-\frac{1}{5}\right)+\frac{8}{3}$$

Whakareatia -\frac{5}{3} ki te -\frac{1}{5} 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.

$$u=\frac{1+8}{3}$$

Tāpiri \frac{8}{3} ki te \frac{1}{3} 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.

$$u=3$$

Kua oti te pūnaha te whakatau.

$$u=3,x=-\frac{1}{5}$$

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.

$$3u+5x=8,5u+5x=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\5&5\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}8\\14\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&5\\5&5\end{matrix}\right))\left(\begin{matrix}3&5\\5&5\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&5\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&5\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&5\end{matrix}\right))\left(\begin{matrix}8\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-5\times 5}&-\frac{5}{3\times 5-5\times 5}\\-\frac{5}{3\times 5-5\times 5}&\frac{3}{3\times 5-5\times 5}\end{matrix}\right)\left(\begin{matrix}8\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}8\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 8+\frac{1}{2}\times 14\\\frac{1}{2}\times 8-\frac{3}{10}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}3\\-\frac{1}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa u me x.

$$u=3,x=-\frac{1}{5}$$

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.

$$3u+5x=8,5u+5x=14$$

Me tango 5u+5x=14 mai i 3u+5x=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3u-5u+5x-5x=8-14$$

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.

$$3u-5u=8-14$$

Tāpiri 3u ki te -5u.

$$-2u=8-14$$

Tāpiri 8 ki te -14.

$$-2u=-6$$

Whakawehea ngā taha e rua ki te -2.

$$u=3$$

Whakaurua te 3 mō u ki 5u+5x=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5\times 3+5x=14$$

Whakareatia 5 ki te 3.

$$15+5x=14$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$5x=-1$$

Whakawehea ngā taha e rua ki te 5.

$$x=-\frac{1}{5}$$

Kua oti te pūnaha te whakatau.

$$u=3,x=-\frac{1}{5}$$