Whakaoti i te {l}{-0.8x+2.3y=3.6}{1.6x-1.2y=6.4}

$$\left. \begin{array} { l } { - 0.8 x + 2.3 y = 3.6 } \\ { 1.6 x - 1.2 y = 6.4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=4$$

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.

$$-0.8x+2.3y=3.6,1.6x-1.2y=6.4$$

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

$$-0.8x+2.3y=3.6$$

Me tango \frac{23y}{10} mai i ngā taha e rua o te whārite.

$$-0.8x=-2.3y+3.6$$

Whakawehea ngā taha e rua o te whārite ki te -0.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-1.25\left(-2.3y+3.6\right)$$

Whakareatia -1.25 ki te -\frac{23y}{10}+3.6.

$$x=2.875y-4.5$$

Whakakapia te \frac{23y}{8}-4.5 mō te x ki tērā atu whārite, 1.6x-1.2y=6.4.

$$1.6\left(2.875y-4.5\right)-1.2y=6.4$$

Whakareatia 1.6 ki te \frac{23y}{8}-4.5.

$$4.6y-7.2-1.2y=6.4$$

Tāpiri \frac{23y}{5} ki te -\frac{6y}{5}.

$$3.4y-7.2=6.4$$

Me tāpiri 7.2 ki ngā taha e rua o te whārite.

$$3.4y=13.6$$

Whakawehea ngā taha e rua o te whārite ki te 3.4, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=2.875y-4.5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2.875\times 4-4.5$$

Whakareatia 2.875 ki te 4.

$$x=\frac{23-9}{2}$$

Tāpiri -4.5 ki te 11.5 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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=4$$

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.

$$-0.8x+2.3y=3.6,1.6x-1.2y=6.4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right))\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right))\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right))\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-0.8&2.3\\1.6&-1.2\end{matrix}\right))\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1.2}{-0.8\left(-1.2\right)-2.3\times 1.6}&-\frac{2.3}{-0.8\left(-1.2\right)-2.3\times 1.6}\\-\frac{1.6}{-0.8\left(-1.2\right)-2.3\times 1.6}&-\frac{0.8}{-0.8\left(-1.2\right)-2.3\times 1.6}\end{matrix}\right)\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{34}&\frac{115}{136}\\\frac{10}{17}&\frac{5}{17}\end{matrix}\right)\left(\begin{matrix}3.6\\6.4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{34}\times 3.6+\frac{115}{136}\times 6.4\\\frac{10}{17}\times 3.6+\frac{5}{17}\times 6.4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=4$$

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.

$$-0.8x+2.3y=3.6,1.6x-1.2y=6.4$$

Kia ōrite ai a -\frac{4x}{5} me \frac{8x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -0.8.

$$1.6\left(-0.8\right)x+1.6\times 2.3y=1.6\times 3.6,-0.8\times 1.6x-0.8\left(-1.2\right)y=-0.8\times 6.4$$

Whakarūnātia.

$$-1.28x+3.68y=5.76,-1.28x+0.96y=-5.12$$

Me tango -1.28x+0.96y=-5.12 mai i -1.28x+3.68y=5.76 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-1.28x+1.28x+3.68y-0.96y=\frac{144+128}{25}$$

Tāpiri -\frac{32x}{25} ki te \frac{32x}{25}. Ka whakakore atu ngā kupu -\frac{32x}{25} me \frac{32x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3.68y-0.96y=\frac{144+128}{25}$$

Tāpiri \frac{92y}{25} ki te -\frac{24y}{25}.

$$2.72y=\frac{144+128}{25}$$

Tāpiri 5.76 ki te 5.12 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.

$$2.72y=10.88$$

Whakawehea ngā taha e rua o te whārite ki te 2.72, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki 1.6x-1.2y=6.4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$1.6x-1.2\times 4=6.4$$

Whakareatia -1.2 ki te 4.

$$1.6x-4.8=6.4$$

Me tāpiri 4.8 ki ngā taha e rua o te whārite.

$$1.6x=11.2$$

Whakawehea ngā taha e rua o te whārite ki te 1.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=4$$