Whakaoti i te {l}{2p+3x=10}{p-x+2=0}

$$\left\{ \begin{array} { l } { 2 p + 3 x = 10 } \\ { p - x + 2 = 0 } \end{array} \right.$$

Whakaoti mō p, x

$$x = \frac{14}{5} = 2\frac{4}{5} = 2.8$$

$$p=\frac{4}{5}=0.8$$

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.

$$2p+3x=10,p-x+2=0$$

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

$$2p+3x=10$$

Me tango 3x mai i ngā taha e rua o te whārite.

$$2p=-3x+10$$

Whakawehea ngā taha e rua ki te 2.

$$p=\frac{1}{2}\left(-3x+10\right)$$

Whakareatia \frac{1}{2} ki te -3x+10.

$$p=-\frac{3}{2}x+5$$

Whakakapia te -\frac{3x}{2}+5 mō te p ki tērā atu whārite, p-x+2=0.

$$-\frac{3}{2}x+5-x+2=0$$

Tāpiri -\frac{3x}{2} ki te -x.

$$-\frac{5}{2}x+5+2=0$$

Tāpiri 5 ki te 2.

$$-\frac{5}{2}x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$-\frac{5}{2}x=-7$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{14}{5}$$

Whakaurua te \frac{14}{5} mō x ki p=-\frac{3}{2}x+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p=-\frac{3}{2}\times \frac{14}{5}+5$$

Whakareatia -\frac{3}{2} ki te \frac{14}{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.

$$p=-\frac{21}{5}+5$$

Tāpiri 5 ki te -\frac{21}{5}.

$$p=\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$p=\frac{4}{5},x=\frac{14}{5}$$

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.

$$2p+3x=10,p-x+2=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&-1\end{matrix}\right)\left(\begin{matrix}p\\x\end{matrix}\right)=\left(\begin{matrix}10\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}2&3\\1&-1\end{matrix}\right)\left(\begin{matrix}p\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}10\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}10\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&-1\end{matrix}\right))\left(\begin{matrix}10\\-2\end{matrix}\right)$$

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{3}{5}\\\frac{1}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}10\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 10+\frac{3}{5}\left(-2\right)\\\frac{1}{5}\times 10-\frac{2}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\\\frac{14}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa p me x.

$$p=\frac{4}{5},x=\frac{14}{5}$$

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.

$$2p+3x=10,p-x+2=0$$

Kia ōrite ai a 2p me p, 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 2.

$$2p+3x=10,2p+2\left(-1\right)x+2\times 2=0$$

Whakarūnātia.

$$2p+3x=10,2p-2x+4=0$$

Me tango 2p-2x+4=0 mai i 2p+3x=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2p-2p+3x+2x-4=10$$

Tāpiri 2p ki te -2p. Ka whakakore atu ngā kupu 2p me -2p, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3x+2x-4=10$$

Tāpiri 3x ki te 2x.

$$5x-4=10$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$5x=14$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{14}{5}$$

Whakaurua te \frac{14}{5} mō x ki p-x+2=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p-\frac{14}{5}+2=0$$

Tāpiri -\frac{14}{5} ki te 2.

$$p-\frac{4}{5}=0$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$p=\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$p=\frac{4}{5},x=\frac{14}{5}$$