Whakaoti i te {l}{44=112k+b}{16=82k+b}

$$\left\{ \begin{array} { l } { 44 = 112 k + b } \\ { 16 = 82 k + b } \end{array} \right.$$

Whakaoti mō k, b

$$k=\frac{14}{15}\approx 0.933333333$$

$$b = -\frac{908}{15} = -60\frac{8}{15} \approx -60.533333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$112k+b=44$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$82k+b=16$$

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.

$$112k+b=44,82k+b=16$$

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

$$112k+b=44$$

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

$$112k=-b+44$$

Whakawehea ngā taha e rua ki te 112.

$$k=\frac{1}{112}\left(-b+44\right)$$

Whakareatia \frac{1}{112} ki te -b+44.

$$k=-\frac{1}{112}b+\frac{11}{28}$$

Whakakapia te -\frac{b}{112}+\frac{11}{28} mō te k ki tērā atu whārite, 82k+b=16.

$$82\left(-\frac{1}{112}b+\frac{11}{28}\right)+b=16$$

Whakareatia 82 ki te -\frac{b}{112}+\frac{11}{28}.

$$-\frac{41}{56}b+\frac{451}{14}+b=16$$

Tāpiri -\frac{41b}{56} ki te b.

$$\frac{15}{56}b+\frac{451}{14}=16$$

Me tango \frac{451}{14} mai i ngā taha e rua o te whārite.

$$\frac{15}{56}b=-\frac{227}{14}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{15}{56}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-\frac{908}{15}$$

Whakaurua te -\frac{908}{15} mō b ki k=-\frac{1}{112}b+\frac{11}{28}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=-\frac{1}{112}\left(-\frac{908}{15}\right)+\frac{11}{28}$$

Whakareatia -\frac{1}{112} ki te -\frac{908}{15} 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.

$$k=\frac{227}{420}+\frac{11}{28}$$

Tāpiri \frac{11}{28} ki te \frac{227}{420} 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.

$$k=\frac{14}{15}$$

Kua oti te pūnaha te whakatau.

$$k=\frac{14}{15},b=-\frac{908}{15}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$112k+b=44$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$82k+b=16$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$112k+b=44,82k+b=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}112&1\\82&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}44\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}112&1\\82&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}112&1\\82&1\end{matrix}\right))\left(\begin{matrix}112&1\\82&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}112&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}112&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}112&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{112-82}&-\frac{1}{112-82}\\-\frac{82}{112-82}&\frac{112}{112-82}\end{matrix}\right)\left(\begin{matrix}44\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}&-\frac{1}{30}\\-\frac{41}{15}&\frac{56}{15}\end{matrix}\right)\left(\begin{matrix}44\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}\times 44-\frac{1}{30}\times 16\\-\frac{41}{15}\times 44+\frac{56}{15}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{14}{15}\\-\frac{908}{15}\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me b.

$$k=\frac{14}{15},b=-\frac{908}{15}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$112k+b=44$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$82k+b=16$$

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.

$$112k+b=44,82k+b=16$$

Me tango 82k+b=16 mai i 112k+b=44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$112k-82k+b-b=44-16$$

Tāpiri b ki te -b. Ka whakakore atu ngā kupu b me -b, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$112k-82k=44-16$$

Tāpiri 112k ki te -82k.

$$30k=44-16$$

Tāpiri 44 ki te -16.

$$30k=28$$

Whakawehea ngā taha e rua ki te 30.

$$k=\frac{14}{15}$$

Whakaurua te \frac{14}{15} mō k ki 82k+b=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$82\times \frac{14}{15}+b=16$$

Whakareatia 82 ki te \frac{14}{15}.

$$\frac{1148}{15}+b=16$$

Me tango \frac{1148}{15} mai i ngā taha e rua o te whārite.

$$b=-\frac{908}{15}$$

Kua oti te pūnaha te whakatau.

$$k=\frac{14}{15},b=-\frac{908}{15}$$