Whakaoti i te {l}{44k+b=72}{48k+b=64}

$$\left\{ \begin{array} { l } { 44 k + b = 72 } \\ { 48 k + b = 64 } \end{array} \right.$$

Whakaoti mō k, b

$$k=-2$$

$$b=160$$

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.

$$44k+b=72,48k+b=64$$

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

$$44k+b=72$$

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

$$44k=-b+72$$

Whakawehea ngā taha e rua ki te 44.

$$k=\frac{1}{44}\left(-b+72\right)$$

Whakareatia \frac{1}{44} ki te -b+72.

$$k=-\frac{1}{44}b+\frac{18}{11}$$

Whakakapia te -\frac{b}{44}+\frac{18}{11} mō te k ki tērā atu whārite, 48k+b=64.

$$48\left(-\frac{1}{44}b+\frac{18}{11}\right)+b=64$$

Whakareatia 48 ki te -\frac{b}{44}+\frac{18}{11}.

$$-\frac{12}{11}b+\frac{864}{11}+b=64$$

Tāpiri -\frac{12b}{11} ki te b.

$$-\frac{1}{11}b+\frac{864}{11}=64$$

Me tango \frac{864}{11} mai i ngā taha e rua o te whārite.

$$-\frac{1}{11}b=-\frac{160}{11}$$

Me whakarea ngā taha e rua ki te -11.

$$b=160$$

Whakaurua te 160 mō b ki k=-\frac{1}{44}b+\frac{18}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=-\frac{1}{44}\times 160+\frac{18}{11}$$

Whakareatia -\frac{1}{44} ki te 160.

$$k=\frac{-40+18}{11}$$

Tāpiri \frac{18}{11} ki te -\frac{40}{11} 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.

$$k=-2$$

Kua oti te pūnaha te whakatau.

$$k=-2,b=160$$

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.

$$44k+b=72,48k+b=64$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}44&1\\48&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}72\\64\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}44&1\\48&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}44&1\\48&1\end{matrix}\right))\left(\begin{matrix}44&1\\48&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}44&1\\48&1\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}44&1\\48&1\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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

$$\left(\begin{matrix}k\\b\end{matrix}\right)=inverse(\left(\begin{matrix}44&1\\48&1\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{44-48}&-\frac{1}{44-48}\\-\frac{48}{44-48}&\frac{44}{44-48}\end{matrix}\right)\left(\begin{matrix}72\\64\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\12&-11\end{matrix}\right)\left(\begin{matrix}72\\64\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 72+\frac{1}{4}\times 64\\12\times 72-11\times 64\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}-2\\160\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me b.

$$k=-2,b=160$$

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.

$$44k+b=72,48k+b=64$$

Me tango 48k+b=64 mai i 44k+b=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$44k-48k+b-b=72-64$$

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.

$$44k-48k=72-64$$

Tāpiri 44k ki te -48k.

$$-4k=72-64$$

Tāpiri 72 ki te -64.

$$-4k=8$$

Whakawehea ngā taha e rua ki te -4.

$$k=-2$$

Whakaurua te -2 mō k ki 48k+b=64. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$48\left(-2\right)+b=64$$

Whakareatia 48 ki te -2.

$$-96+b=64$$

Me tāpiri 96 ki ngā taha e rua o te whārite.

$$b=160$$

Kua oti te pūnaha te whakatau.

$$k=-2,b=160$$