Whakaoti i te {l}{0.07r+0.02t=0.16}{0.05r-0.03t=0.21}

$$\left\{ \begin{array} { l } { 0.07 r + 0.02 t = 0.16 } \\ { 0.05 r - 0.03 t = 0.21 } \end{array} \right.$$

Whakaoti mō r, t

$$r = \frac{90}{31} = 2\frac{28}{31} \approx 2.903225806$$

$$t = -\frac{67}{31} = -2\frac{5}{31} \approx -2.161290323$$

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.

$$0.07r+0.02t=0.16,0.05r-0.03t=0.21$$

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

$$0.07r+0.02t=0.16$$

Me tango \frac{t}{50} mai i ngā taha e rua o te whārite.

$$0.07r=-0.02t+0.16$$

Whakawehea ngā taha e rua o te whārite ki te 0.07, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$r=\frac{100}{7}\left(-0.02t+0.16\right)$$

Whakareatia \frac{100}{7} ki te -\frac{t}{50}+0.16.

$$r=-\frac{2}{7}t+\frac{16}{7}$$

Whakakapia te \frac{-2t+16}{7} mō te r ki tērā atu whārite, 0.05r-0.03t=0.21.

$$0.05\left(-\frac{2}{7}t+\frac{16}{7}\right)-0.03t=0.21$$

Whakareatia 0.05 ki te \frac{-2t+16}{7}.

$$-\frac{1}{70}t+\frac{4}{35}-0.03t=0.21$$

Tāpiri -\frac{t}{70} ki te -\frac{3t}{100}.

$$-\frac{31}{700}t+\frac{4}{35}=0.21$$

Me tango \frac{4}{35} mai i ngā taha e rua o te whārite.

$$-\frac{31}{700}t=\frac{67}{700}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{31}{700}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$t=-\frac{67}{31}$$

Whakaurua te -\frac{67}{31} mō t ki r=-\frac{2}{7}t+\frac{16}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō r hāngai tonu.

$$r=-\frac{2}{7}\left(-\frac{67}{31}\right)+\frac{16}{7}$$

Whakareatia -\frac{2}{7} ki te -\frac{67}{31} 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.

$$r=\frac{134}{217}+\frac{16}{7}$$

Tāpiri \frac{16}{7} ki te \frac{134}{217} 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.

$$r=\frac{90}{31}$$

Kua oti te pūnaha te whakatau.

$$r=\frac{90}{31},t=-\frac{67}{31}$$

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.

$$0.07r+0.02t=0.16,0.05r-0.03t=0.21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right)\left(\begin{matrix}r\\t\end{matrix}\right)=\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right))\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right)\left(\begin{matrix}r\\t\end{matrix}\right)=inverse(\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right))\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}r\\t\end{matrix}\right)=inverse(\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right))\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

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

$$\left(\begin{matrix}r\\t\end{matrix}\right)=inverse(\left(\begin{matrix}0.07&0.02\\0.05&-0.03\end{matrix}\right))\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}r\\t\end{matrix}\right)=\left(\begin{matrix}-\frac{0.03}{0.07\left(-0.03\right)-0.02\times 0.05}&-\frac{0.02}{0.07\left(-0.03\right)-0.02\times 0.05}\\-\frac{0.05}{0.07\left(-0.03\right)-0.02\times 0.05}&\frac{0.07}{0.07\left(-0.03\right)-0.02\times 0.05}\end{matrix}\right)\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}r\\t\end{matrix}\right)=\left(\begin{matrix}\frac{300}{31}&\frac{200}{31}\\\frac{500}{31}&-\frac{700}{31}\end{matrix}\right)\left(\begin{matrix}0.16\\0.21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}r\\t\end{matrix}\right)=\left(\begin{matrix}\frac{300}{31}\times 0.16+\frac{200}{31}\times 0.21\\\frac{500}{31}\times 0.16-\frac{700}{31}\times 0.21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}r\\t\end{matrix}\right)=\left(\begin{matrix}\frac{90}{31}\\-\frac{67}{31}\end{matrix}\right)$$

Tangohia ngā huānga poukapa r me t.

$$r=\frac{90}{31},t=-\frac{67}{31}$$

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.

$$0.07r+0.02t=0.16,0.05r-0.03t=0.21$$

Kia ōrite ai a \frac{7r}{100} me \frac{r}{20}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.05 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.07.

$$0.05\times 0.07r+0.05\times 0.02t=0.05\times 0.16,0.07\times 0.05r+0.07\left(-0.03\right)t=0.07\times 0.21$$

Whakarūnātia.

$$0.0035r+0.001t=0.008,0.0035r-0.0021t=0.0147$$

Me tango 0.0035r-0.0021t=0.0147 mai i 0.0035r+0.001t=0.008 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.0035r-0.0035r+0.001t+0.0021t=0.008-0.0147$$

Tāpiri \frac{7r}{2000} ki te -\frac{7r}{2000}. Ka whakakore atu ngā kupu \frac{7r}{2000} me -\frac{7r}{2000}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.001t+0.0021t=0.008-0.0147$$

Tāpiri \frac{t}{1000} ki te \frac{21t}{10000}.

$$0.0031t=0.008-0.0147$$

Tāpiri 0.008 ki te -0.0147 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.

$$0.0031t=-0.0067$$

Whakawehea ngā taha e rua o te whārite ki te 0.0031, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$t=-\frac{67}{31}$$

Whakaurua te -\frac{67}{31} mō t ki 0.05r-0.03t=0.21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō r hāngai tonu.

$$0.05r-0.03\left(-\frac{67}{31}\right)=0.21$$

Whakareatia -0.03 ki te -\frac{67}{31} 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.

$$0.05r+\frac{201}{3100}=0.21$$

Me tango \frac{201}{3100} mai i ngā taha e rua o te whārite.

$$0.05r=\frac{9}{62}$$

Me whakarea ngā taha e rua ki te 20.

$$r=\frac{90}{31}$$

Kua oti te pūnaha te whakatau.

$$r=\frac{90}{31},t=-\frac{67}{31}$$