Whakaoti i te {c}{3x+y=5}{-2x+2y=7}

$$\left. \begin{array} { c } { 3 x + y = 5 } \\ { - 2 x + 2 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3}{8}=0.375$$

$$y = \frac{31}{8} = 3\frac{7}{8} = 3.875$$

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.

$$3x+y=5,-2x+2y=7$$

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

$$3x+y=5$$

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

$$3x=-y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y+\frac{5}{3}$$

Whakakapia te \frac{-y+5}{3} mō te x ki tērā atu whārite, -2x+2y=7.

$$-2\left(-\frac{1}{3}y+\frac{5}{3}\right)+2y=7$$

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

$$\frac{2}{3}y-\frac{10}{3}+2y=7$$

Tāpiri \frac{2y}{3} ki te 2y.

$$\frac{8}{3}y-\frac{10}{3}=7$$

Me tāpiri \frac{10}{3} ki ngā taha e rua o te whārite.

$$\frac{8}{3}y=\frac{31}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{31}{8}$$

Whakaurua te \frac{31}{8} mō y ki x=-\frac{1}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{31}{8}+\frac{5}{3}$$

Whakareatia -\frac{1}{3} ki te \frac{31}{8} 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.

$$x=-\frac{31}{24}+\frac{5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{31}{24} 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.

$$x=\frac{3}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{8},y=\frac{31}{8}$$

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.

$$3x+y=5,-2x+2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\-2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\-2&2\end{matrix}\right))\left(\begin{matrix}3&1\\-2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-2&2\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-2&2\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\-2&2\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-2\right)}&-\frac{1}{3\times 2-\left(-2\right)}\\-\frac{-2}{3\times 2-\left(-2\right)}&\frac{3}{3\times 2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{8}\\\frac{1}{4}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 5-\frac{1}{8}\times 7\\\frac{1}{4}\times 5+\frac{3}{8}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\\\frac{31}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{8},y=\frac{31}{8}$$

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.

$$3x+y=5,-2x+2y=7$$

Kia ōrite ai a 3x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-2\times 3x-2y=-2\times 5,3\left(-2\right)x+3\times 2y=3\times 7$$

Whakarūnātia.

$$-6x-2y=-10,-6x+6y=21$$

Me tango -6x+6y=21 mai i -6x-2y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-2y-6y=-10-21$$

Tāpiri -6x ki te 6x. Ka whakakore atu ngā kupu -6x me 6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-6y=-10-21$$

Tāpiri -2y ki te -6y.

$$-8y=-10-21$$

Tāpiri -10 ki te -21.

$$-8y=-31$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{31}{8}$$

Whakaurua te \frac{31}{8} mō y ki -2x+2y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+2\times \frac{31}{8}=7$$

Whakareatia 2 ki te \frac{31}{8}.

$$-2x+\frac{31}{4}=7$$

Me tango \frac{31}{4} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{3}{4}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{3}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{8},y=\frac{31}{8}$$