Whakaoti i te {l}{p+b=130}{p+1.09b=136.75}

$$\left. \begin{array} { l } { p + b = 130 } \\ { p + 1.09 b = 136.75 } \end{array} \right.$$

Whakaoti mō p, b

$$p=55$$

$$b=75$$

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.

$$p+b=130,p+1.09b=136.75$$

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

$$p+b=130$$

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

$$p=-b+130$$

Whakakapia te -b+130 mō te p ki tērā atu whārite, p+1.09b=136.75.

$$-b+130+1.09b=136.75$$

Tāpiri -b ki te \frac{109b}{100}.

$$0.09b+130=136.75$$

Me tango 130 mai i ngā taha e rua o te whārite.

$$0.09b=6.75$$

Whakawehea ngā taha e rua o te whārite ki te 0.09, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=75$$

Whakaurua te 75 mō b ki p=-b+130. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p=-75+130$$

Tāpiri 130 ki te -75.

$$p=55$$

Kua oti te pūnaha te whakatau.

$$p=55,b=75$$

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.

$$p+b=130,p+1.09b=136.75$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&1.09\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right))\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right))\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right))\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&1.09\end{matrix}\right))\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1.09}{1.09-1}&-\frac{1}{1.09-1}\\-\frac{1}{1.09-1}&\frac{1}{1.09-1}\end{matrix}\right)\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}\frac{109}{9}&-\frac{100}{9}\\-\frac{100}{9}&\frac{100}{9}\end{matrix}\right)\left(\begin{matrix}130\\136.75\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}\frac{109}{9}\times 130-\frac{100}{9}\times 136.75\\-\frac{100}{9}\times 130+\frac{100}{9}\times 136.75\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}55\\75\end{matrix}\right)$$

Tangohia ngā huānga poukapa p me b.

$$p=55,b=75$$

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.

$$p+b=130,p+1.09b=136.75$$

Me tango p+1.09b=136.75 mai i p+b=130 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$p-p+b-1.09b=130-136.75$$

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

$$b-1.09b=130-136.75$$

Tāpiri b ki te -\frac{109b}{100}.

$$-0.09b=130-136.75$$

Tāpiri 130 ki te -136.75.

$$-0.09b=-6.75$$

Whakawehea ngā taha e rua o te whārite ki te -0.09, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=75$$

Whakaurua te 75 mō b ki p+1.09b=136.75. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p+1.09\times 75=136.75$$

Whakareatia 1.09 ki te 75.

$$p+81.75=136.75$$

Me tango 81.75 mai i ngā taha e rua o te whārite.

$$p=55$$

Kua oti te pūnaha te whakatau.

$$p=55,b=75$$