Whakaoti i te {l}{3x+2y=1.714}{2x+y=1}

$$\left. \begin{array} { l } { 3 x + 2 y = 1.714 } \\ { 2 x + y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0.286$$

$$y=0.428$$

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.

$$3x+2y=1.714,2x+y=1$$

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

$$3x+2y=1.714$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+1.714$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+1.714\right)$$

Whakareatia \frac{1}{3} ki te -2y+1.714.

$$x=-\frac{2}{3}y+\frac{857}{1500}$$

Whakakapia te -\frac{2y}{3}+\frac{857}{1500} mō te x ki tērā atu whārite, 2x+y=1.

$$2\left(-\frac{2}{3}y+\frac{857}{1500}\right)+y=1$$

Whakareatia 2 ki te -\frac{2y}{3}+\frac{857}{1500}.

$$-\frac{4}{3}y+\frac{857}{750}+y=1$$

Tāpiri -\frac{4y}{3} ki te y.

$$-\frac{1}{3}y+\frac{857}{750}=1$$

Me tango \frac{857}{750} mai i ngā taha e rua o te whārite.

$$-\frac{1}{3}y=-\frac{107}{750}$$

Me whakarea ngā taha e rua ki te -3.

$$y=\frac{107}{250}$$

Whakaurua te \frac{107}{250} mō y ki x=-\frac{2}{3}y+\frac{857}{1500}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{107}{250}+\frac{857}{1500}$$

Whakareatia -\frac{2}{3} ki te \frac{107}{250} 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.

$$x=-\frac{107}{375}+\frac{857}{1500}$$

Tāpiri \frac{857}{1500} ki te -\frac{107}{375} 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.

$$x=\frac{143}{500}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{143}{500},y=\frac{107}{250}$$

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.

$$3x+2y=1.714,2x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\2&1\end{matrix}\right))\left(\begin{matrix}3&2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&1\end{matrix}\right))\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&1\end{matrix}\right))\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&1\end{matrix}\right))\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-2\times 2}&-\frac{2}{3-2\times 2}\\-\frac{2}{3-2\times 2}&\frac{3}{3-2\times 2}\end{matrix}\right)\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&2\\2&-3\end{matrix}\right)\left(\begin{matrix}1.714\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1.714+2\\2\times 1.714-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0.286\\0.428\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0.286,y=0.428$$

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.

$$3x+2y=1.714,2x+y=1$$

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.

$$2\times 3x+2\times 2y=2\times 1.714,3\times 2x+3y=3$$

Whakarūnātia.

$$6x+4y=3.428,6x+3y=3$$

Me tango 6x+3y=3 mai i 6x+4y=3.428 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+4y-3y=3.428-3$$

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.

$$4y-3y=3.428-3$$

Tāpiri 4y ki te -3y.

$$y=3.428-3$$

Tāpiri 3.428 ki te -3.

$$y=0.428$$

Whakaurua te 0.428 mō y ki 2x+y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+0.428=1$$

Me tango 0.428 mai i ngā taha e rua o te whārite.

$$2x=0.572$$

Whakawehea ngā taha e rua ki te 2.

$$x=0.286$$

Kua oti te pūnaha te whakatau.

$$x=0.286,y=0.428$$