Whakaoti i te 3x+5y=8text{and}x-2y=-1

$$3 x + 5 y = 8 \text { and } x - 2 y = - 1$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

$$3x+5y=8,x-2y=-1$$

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

$$3x+5y=8$$

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

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{-5y+8}{3} mō te x ki tērā atu whārite, x-2y=-1.

$$-\frac{5}{3}y+\frac{8}{3}-2y=-1$$

Tāpiri -\frac{5y}{3} ki te -2y.

$$-\frac{11}{3}y+\frac{8}{3}=-1$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-\frac{11}{3}y=-\frac{11}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

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

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

Tāpiri \frac{8}{3} ki te -\frac{5}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

$$3x+5y=8,x-2y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\1&-2\end{matrix}\right))\left(\begin{matrix}3&5\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-2\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-5}&-\frac{5}{3\left(-2\right)-5}\\-\frac{1}{3\left(-2\right)-5}&\frac{3}{3\left(-2\right)-5}\end{matrix}\right)\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{5}{11}\\\frac{1}{11}&-\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 8+\frac{5}{11}\left(-1\right)\\\frac{1}{11}\times 8-\frac{3}{11}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

$$3x+5y=8,x-2y=-1$$

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.

$$3x+5y=8,3x+3\left(-2\right)y=3\left(-1\right)$$

Whakarūnātia.

$$3x+5y=8,3x-6y=-3$$

Me tango 3x-6y=-3 mai i 3x+5y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+5y+6y=8+3$$

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.

$$5y+6y=8+3$$

Tāpiri 5y ki te 6y.

$$11y=8+3$$

Tāpiri 8 ki te 3.

$$11y=11$$

Whakawehea ngā taha e rua ki te 11.

$$y=1$$

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

$$x-2=-1$$

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

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$