Whakaoti i te {r}{80=a+5*b}{a+b=49}

$$\left.\begin{array} { r } { 80 = a + 5 \cdot b } \\ { a + b = 49 } \end{array} \right\}$$

Whakaoti mō a, b

$$a = \frac{165}{4} = 41\frac{1}{4} = 41.25$$

$$b = \frac{31}{4} = 7\frac{3}{4} = 7.75$$

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.

$$a+5b=80$$

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.

$$a+5b=80,a+b=49$$

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

$$a+5b=80$$

Me tango 5b mai i ngā taha e rua o te whārite.

$$a=-5b+80$$

Whakakapia te -5b+80 mō te a ki tērā atu whārite, a+b=49.

$$-5b+80+b=49$$

Tāpiri -5b ki te b.

$$-4b+80=49$$

Me tango 80 mai i ngā taha e rua o te whārite.

$$-4b=-31$$

Whakawehea ngā taha e rua ki te -4.

$$b=\frac{31}{4}$$

Whakaurua te \frac{31}{4} mō b ki a=-5b+80. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-5\times \frac{31}{4}+80$$

Whakareatia -5 ki te \frac{31}{4}.

$$a=-\frac{155}{4}+80$$

Tāpiri 80 ki te -\frac{155}{4}.

$$a=\frac{165}{4}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{165}{4},b=\frac{31}{4}$$

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.

$$a+5b=80$$

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

$$a+5b=80,a+b=49$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&5\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}80\\49\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&5\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&5\\1&1\end{matrix}\right))\left(\begin{matrix}1&5\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&1\end{matrix}\right))\left(\begin{matrix}80\\49\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&1\end{matrix}\right))\left(\begin{matrix}80\\49\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&1\end{matrix}\right))\left(\begin{matrix}80\\49\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-5}&-\frac{5}{1-5}\\-\frac{1}{1-5}&\frac{1}{1-5}\end{matrix}\right)\left(\begin{matrix}80\\49\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{5}{4}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}80\\49\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 80+\frac{5}{4}\times 49\\\frac{1}{4}\times 80-\frac{1}{4}\times 49\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{165}{4}\\\frac{31}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{165}{4},b=\frac{31}{4}$$

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.

$$a+5b=80$$

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.

$$a+5b=80,a+b=49$$

Me tango a+b=49 mai i a+5b=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+5b-b=80-49$$

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

$$5b-b=80-49$$

Tāpiri 5b ki te -b.

$$4b=80-49$$

Tāpiri 80 ki te -49.

$$4b=31$$

Whakawehea ngā taha e rua ki te 4.

$$b=\frac{31}{4}$$

Whakaurua te \frac{31}{4} mō b ki a+b=49. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a+\frac{31}{4}=49$$

Me tango \frac{31}{4} mai i ngā taha e rua o te whārite.

$$a=\frac{165}{4}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{165}{4},b=\frac{31}{4}$$