Whakaoti i te {l}{0.3x+y=4.8}{x-y=11}

$$\left\{ \begin{array} { l } { 0.3 x + y = 4.8 } \\ { x - y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{158}{13} = 12\frac{2}{13} \approx 12.153846154$$

$$y = \frac{15}{13} = 1\frac{2}{13} \approx 1.153846154$$

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.

$$0.3x+y=4.8,x-y=11$$

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

$$0.3x+y=4.8$$

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

$$0.3x=-y+4.8$$

Whakawehea ngā taha e rua o te whārite ki te 0.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{10}{3}\left(-y+4.8\right)$$

Whakareatia \frac{10}{3} ki te -y+4.8.

$$x=-\frac{10}{3}y+16$$

Whakakapia te -\frac{10y}{3}+16 mō te x ki tērā atu whārite, x-y=11.

$$-\frac{10}{3}y+16-y=11$$

Tāpiri -\frac{10y}{3} ki te -y.

$$-\frac{13}{3}y+16=11$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-\frac{13}{3}y=-5$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{15}{13}$$

Whakaurua te \frac{15}{13} mō y ki x=-\frac{10}{3}y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{10}{3}\times \frac{15}{13}+16$$

Whakareatia -\frac{10}{3} ki te \frac{15}{13} 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.

$$x=-\frac{50}{13}+16$$

Tāpiri 16 ki te -\frac{50}{13}.

$$x=\frac{158}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{158}{13},y=\frac{15}{13}$$

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.

$$0.3x+y=4.8,x-y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{0.3\left(-1\right)-1}&-\frac{1}{0.3\left(-1\right)-1}\\-\frac{1}{0.3\left(-1\right)-1}&\frac{0.3}{0.3\left(-1\right)-1}\end{matrix}\right)\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{13}&\frac{10}{13}\\\frac{10}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}4.8\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{13}\times 4.8+\frac{10}{13}\times 11\\\frac{10}{13}\times 4.8-\frac{3}{13}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{158}{13}\\\frac{15}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{158}{13},y=\frac{15}{13}$$

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.

$$0.3x+y=4.8,x-y=11$$

Kia ōrite ai a \frac{3x}{10} 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 0.3.

$$0.3x+y=4.8,0.3x+0.3\left(-1\right)y=0.3\times 11$$

Whakarūnātia.

$$0.3x+y=4.8,0.3x-0.3y=3.3$$

Me tango 0.3x-0.3y=3.3 mai i 0.3x+y=4.8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.3x-0.3x+y+0.3y=4.8-3.3$$

Tāpiri \frac{3x}{10} ki te -\frac{3x}{10}. Ka whakakore atu ngā kupu \frac{3x}{10} me -\frac{3x}{10}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y+0.3y=4.8-3.3$$

Tāpiri y ki te \frac{3y}{10}.

$$1.3y=4.8-3.3$$

Tāpiri 4.8 ki te -3.3 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.

$$1.3y=1.5$$

Whakawehea ngā taha e rua o te whārite ki te 1.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{15}{13}$$

Whakaurua te \frac{15}{13} mō y ki x-y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{15}{13}=11$$

Me tāpiri \frac{15}{13} ki ngā taha e rua o te whārite.

$$x=\frac{158}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{158}{13},y=\frac{15}{13}$$