Whakaoti i te {c}{x+y=64}{0.12x-0.26y=0.19}

$$\left. \begin{array} { c } { x + y = 64 } \\ { 0.12 x - 0.26 y = 0.19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{1683}{38} = 44\frac{11}{38} \approx 44.289473684$$

$$y = \frac{749}{38} = 19\frac{27}{38} \approx 19.710526316$$

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.

$$x+y=64,0.12x-0.26y=0.19$$

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

$$x+y=64$$

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

$$x=-y+64$$

Whakakapia te -y+64 mō te x ki tērā atu whārite, 0.12x-0.26y=0.19.

$$0.12\left(-y+64\right)-0.26y=0.19$$

Whakareatia 0.12 ki te -y+64.

$$-0.12y+7.68-0.26y=0.19$$

Tāpiri -\frac{3y}{25} ki te -\frac{13y}{50}.

$$-0.38y+7.68=0.19$$

Me tango 7.68 mai i ngā taha e rua o te whārite.

$$-0.38y=-7.49$$

Whakawehea ngā taha e rua o te whārite ki te -0.38, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{749}{38}$$

Whakaurua te \frac{749}{38} mō y ki x=-y+64. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{749}{38}+64$$

Tāpiri 64 ki te -\frac{749}{38}.

$$x=\frac{1683}{38}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1683}{38},y=\frac{749}{38}$$

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.

$$x+y=64,0.12x-0.26y=0.19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right))\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right))\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right))\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&-0.26\end{matrix}\right))\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.26}{-0.26-0.12}&-\frac{1}{-0.26-0.12}\\-\frac{0.12}{-0.26-0.12}&\frac{1}{-0.26-0.12}\end{matrix}\right)\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{19}&\frac{50}{19}\\\frac{6}{19}&-\frac{50}{19}\end{matrix}\right)\left(\begin{matrix}64\\0.19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{19}\times 64+\frac{50}{19}\times 0.19\\\frac{6}{19}\times 64-\frac{50}{19}\times 0.19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1683}{38}\\\frac{749}{38}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1683}{38},y=\frac{749}{38}$$

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.

$$x+y=64,0.12x-0.26y=0.19$$

Kia ōrite ai a x me \frac{3x}{25}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$0.12x+0.12y=0.12\times 64,0.12x-0.26y=0.19$$

Whakarūnātia.

$$0.12x+0.12y=7.68,0.12x-0.26y=0.19$$

Me tango 0.12x-0.26y=0.19 mai i 0.12x+0.12y=7.68 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.12x-0.12x+0.12y+0.26y=7.68-0.19$$

Tāpiri \frac{3x}{25} ki te -\frac{3x}{25}. Ka whakakore atu ngā kupu \frac{3x}{25} me -\frac{3x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.12y+0.26y=7.68-0.19$$

Tāpiri \frac{3y}{25} ki te \frac{13y}{50}.

$$0.38y=7.68-0.19$$

Tāpiri 7.68 ki te -0.19 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.

$$0.38y=7.49$$

Whakawehea ngā taha e rua o te whārite ki te 0.38, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{749}{38}$$

Whakaurua te \frac{749}{38} mō y ki 0.12x-0.26y=0.19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.12x-0.26\times \frac{749}{38}=0.19$$

Whakareatia -0.26 ki te \frac{749}{38} 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.

$$0.12x-\frac{9737}{1900}=0.19$$

Me tāpiri \frac{9737}{1900} ki ngā taha e rua o te whārite.

$$0.12x=\frac{5049}{950}$$

Whakawehea ngā taha e rua o te whārite ki te 0.12, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{1683}{38}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1683}{38},y=\frac{749}{38}$$