Whakaoti i te {l}{3x+5y=1}{x+y=2}

$$\left. \begin{array} { l } { 3 x + 5 y = 1 } \\ { x + y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$y = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

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.

$$3x+5y=1,x+y=2$$

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

$$3x+5y=1$$

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

$$3x=-5y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{-5y+1}{3} mō te x ki tērā atu whārite, x+y=2.

$$-\frac{5}{3}y+\frac{1}{3}+y=2$$

Tāpiri -\frac{5y}{3} ki te y.

$$-\frac{2}{3}y+\frac{1}{3}=2$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te -\frac{5}{2} mō y ki x=-\frac{5}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{25}{6}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{25}{6} 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.

$$x=\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{2},y=-\frac{5}{2}$$

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.

$$3x+5y=1,x+y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\-\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{2},y=-\frac{5}{2}$$

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.

$$3x+5y=1,x+y=2$$

Kia ōrite ai a 3x 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 3.

$$3x+5y=1,3x+3y=3\times 2$$

Whakarūnātia.

$$3x+5y=1,3x+3y=6$$

Me tango 3x+3y=6 mai i 3x+5y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+5y-3y=1-6$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-3y=1-6$$

Tāpiri 5y ki te -3y.

$$2y=1-6$$

Tāpiri 1 ki te -6.

$$2y=-5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{2},y=-\frac{5}{2}$$