Whakaoti i te {l}{5x+(3y-2)=0}{x+y=10}

$$\left. \begin{array} { l } { 5 x + ( 3 y - 2 ) = 0 } \\ { x + y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-14$$

$$y=24$$

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.

$$5x+3y-2=0,x+y=10$$

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

$$5x+3y-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$5x+3y=2$$

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

$$5x=-3y+2$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+2\right)$$

Whakareatia \frac{1}{5} ki te -3y+2.

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

Whakakapia te \frac{-3y+2}{5} mō te x ki tērā atu whārite, x+y=10.

$$-\frac{3}{5}y+\frac{2}{5}+y=10$$

Tāpiri -\frac{3y}{5} ki te y.

$$\frac{2}{5}y+\frac{2}{5}=10$$

Me tango \frac{2}{5} mai i ngā taha e rua o te whārite.

$$\frac{2}{5}y=\frac{48}{5}$$

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

$$y=24$$

Whakaurua te 24 mō y ki x=-\frac{3}{5}y+\frac{2}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\times 24+\frac{2}{5}$$

Whakareatia -\frac{3}{5} ki te 24.

$$x=\frac{-72+2}{5}$$

Tāpiri \frac{2}{5} ki te -\frac{72}{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.

$$x=-14$$

Kua oti te pūnaha te whakatau.

$$x=-14,y=24$$

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.

$$5x+3y-2=0,x+y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

$$inverse(\left(\begin{matrix}5&3\\1&1\end{matrix}\right))\left(\begin{matrix}5&3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\1&1\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\1&1\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\1&1\end{matrix}\right))\left(\begin{matrix}2\\10\end{matrix}\right)$$

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\24\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-14,y=24$$

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.

$$5x+3y-2=0,x+y=10$$

Kia ōrite ai a 5x 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 5.

$$5x+3y-2=0,5x+5y=5\times 10$$

Whakarūnātia.

$$5x+3y-2=0,5x+5y=50$$

Me tango 5x+5y=50 mai i 5x+3y-2=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+3y-5y-2=-50$$

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.

$$3y-5y-2=-50$$

Tāpiri 3y ki te -5y.

$$-2y-2=-50$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-2y=-48$$

Whakawehea ngā taha e rua ki te -2.

$$y=24$$

Whakaurua te 24 mō y ki x+y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+24=10$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$x=-14$$

Kua oti te pūnaha te whakatau.

$$x=-14,y=24$$