Whakaoti i te {l}{44=12k+b}{16=82k+b}

$$\left\{ \begin{array} { l } { 44 = 12 k + b } \\ { 16 = 82 k + b } \end{array} \right.$$

Whakaoti mō k, b

$$k=-\frac{2}{5}=-0.4$$

$$b = \frac{244}{5} = 48\frac{4}{5} = 48.8$$

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.

$$12k+b=44$$

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.

$$12k+b=44,82k+b=16$$

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

$$12k+b=44$$

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

$$12k=-b+44$$

Whakawehea ngā taha e rua ki te 12.

$$k=\frac{1}{12}\left(-b+44\right)$$

Whakareatia \frac{1}{12} ki te -b+44.

$$k=-\frac{1}{12}b+\frac{11}{3}$$

Whakakapia te -\frac{b}{12}+\frac{11}{3} mō te k ki tērā atu whārite, 82k+b=16.

$$82\left(-\frac{1}{12}b+\frac{11}{3}\right)+b=16$$

Whakareatia 82 ki te -\frac{b}{12}+\frac{11}{3}.

$$-\frac{41}{6}b+\frac{902}{3}+b=16$$

Tāpiri -\frac{41b}{6} ki te b.

$$-\frac{35}{6}b+\frac{902}{3}=16$$

Me tango \frac{902}{3} mai i ngā taha e rua o te whārite.

$$-\frac{35}{6}b=-\frac{854}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{35}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{244}{5}$$

Whakaurua te \frac{244}{5} mō b ki k=-\frac{1}{12}b+\frac{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=-\frac{1}{12}\times \frac{244}{5}+\frac{11}{3}$$

Whakareatia -\frac{1}{12} ki te \frac{244}{5} 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.

$$k=-\frac{61}{15}+\frac{11}{3}$$

Tāpiri \frac{11}{3} ki te -\frac{61}{15} 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.

$$k=-\frac{2}{5}$$

Kua oti te pūnaha te whakatau.

$$k=-\frac{2}{5},b=\frac{244}{5}$$

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.

$$12k+b=44$$

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.

$$12k+b=44,82k+b=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12&1\\82&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}44\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12&1\\82&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&1\\82&1\end{matrix}\right))\left(\begin{matrix}12&1\\82&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\82&1\end{matrix}\right))\left(\begin{matrix}44\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12-82}&-\frac{1}{12-82}\\-\frac{82}{12-82}&\frac{12}{12-82}\end{matrix}\right)\left(\begin{matrix}44\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{70}&\frac{1}{70}\\\frac{41}{35}&-\frac{6}{35}\end{matrix}\right)\left(\begin{matrix}44\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{70}\times 44+\frac{1}{70}\times 16\\\frac{41}{35}\times 44-\frac{6}{35}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\\\frac{244}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me b.

$$k=-\frac{2}{5},b=\frac{244}{5}$$

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.

$$12k+b=44$$

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.

$$12k+b=44,82k+b=16$$

Me tango 82k+b=16 mai i 12k+b=44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12k-82k+b-b=44-16$$

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.

$$12k-82k=44-16$$

Tāpiri 12k ki te -82k.

$$-70k=44-16$$

Tāpiri 44 ki te -16.

$$-70k=28$$

Whakawehea ngā taha e rua ki te -70.

$$k=-\frac{2}{5}$$

Whakaurua te -\frac{2}{5} 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.

$$82\left(-\frac{2}{5}\right)+b=16$$

Whakareatia 82 ki te -\frac{2}{5}.

$$-\frac{164}{5}+b=16$$

Me tāpiri \frac{164}{5} ki ngā taha e rua o te whārite.

$$b=\frac{244}{5}$$

Kua oti te pūnaha te whakatau.

$$k=-\frac{2}{5},b=\frac{244}{5}$$