Whakaoti i te {l}{y=7.50p+45}{y=-0.60p+300}

$$\left. \begin{array} { l } { y = 7.50 p + 45 } \\ { y = - 0.60 p + 300 } \end{array} \right.$$

Whakaoti mō y, p

$$y = \frac{2530}{9} = 281\frac{1}{9} \approx 281.111111111$$

$$p = \frac{850}{27} = 31\frac{13}{27} \approx 31.481481481$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 7.5p mai i ngā taha e rua.

$$y-7.5p=45$$

Whakaarohia te whārite tuarua. Me tāpiri te 0.6p ki ngā taha e rua.

$$y+0.6p=300$$

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.

$$y-7.5p=45,y+0.6p=300$$

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

$$y-7.5p=45$$

Me tāpiri \frac{15p}{2} ki ngā taha e rua o te whārite.

$$y=7.5p+45$$

Whakakapia te \frac{15p}{2}+45 mō te y ki tērā atu whārite, y+0.6p=300.

$$7.5p+45+0.6p=300$$

Tāpiri \frac{15p}{2} ki te \frac{3p}{5}.

$$8.1p+45=300$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$8.1p=255$$

Whakawehea ngā taha e rua o te whārite ki te 8.1, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$p=\frac{850}{27}$$

Whakaurua te \frac{850}{27} mō p ki y=7.5p+45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=7.5\times \frac{850}{27}+45$$

Whakareatia 7.5 ki te \frac{850}{27} 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.

$$y=\frac{2125}{9}+45$$

Tāpiri 45 ki te \frac{2125}{9}.

$$y=\frac{2530}{9}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{2530}{9},p=\frac{850}{27}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 7.5p mai i ngā taha e rua.

$$y-7.5p=45$$

Whakaarohia te whārite tuarua. Me tāpiri te 0.6p ki ngā taha e rua.

$$y+0.6p=300$$

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

$$y-7.5p=45,y+0.6p=300$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right)\left(\begin{matrix}y\\p\end{matrix}\right)=\left(\begin{matrix}45\\300\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right))\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right)\left(\begin{matrix}y\\p\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right))\left(\begin{matrix}45\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\p\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right))\left(\begin{matrix}45\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\p\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7.5\\1&0.6\end{matrix}\right))\left(\begin{matrix}45\\300\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\p\end{matrix}\right)=\left(\begin{matrix}\frac{0.6}{0.6-\left(-7.5\right)}&-\frac{-7.5}{0.6-\left(-7.5\right)}\\-\frac{1}{0.6-\left(-7.5\right)}&\frac{1}{0.6-\left(-7.5\right)}\end{matrix}\right)\left(\begin{matrix}45\\300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\p\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}&\frac{25}{27}\\-\frac{10}{81}&\frac{10}{81}\end{matrix}\right)\left(\begin{matrix}45\\300\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\p\end{matrix}\right)=\left(\begin{matrix}\frac{2}{27}\times 45+\frac{25}{27}\times 300\\-\frac{10}{81}\times 45+\frac{10}{81}\times 300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\p\end{matrix}\right)=\left(\begin{matrix}\frac{2530}{9}\\\frac{850}{27}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me p.

$$y=\frac{2530}{9},p=\frac{850}{27}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 7.5p mai i ngā taha e rua.

$$y-7.5p=45$$

Whakaarohia te whārite tuarua. Me tāpiri te 0.6p ki ngā taha e rua.

$$y+0.6p=300$$

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.

$$y-7.5p=45,y+0.6p=300$$

Me tango y+0.6p=300 mai i y-7.5p=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-7.5p-0.6p=45-300$$

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

$$-7.5p-0.6p=45-300$$

Tāpiri -\frac{15p}{2} ki te -\frac{3p}{5}.

$$-8.1p=45-300$$

Tāpiri 45 ki te -300.

$$-8.1p=-255$$

Whakawehea ngā taha e rua o te whārite ki te -8.1, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$p=\frac{850}{27}$$

Whakaurua te \frac{850}{27} mō p ki y+0.6p=300. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+0.6\times \frac{850}{27}=300$$

Whakareatia 0.6 ki te \frac{850}{27} 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.

$$y+\frac{170}{9}=300$$

Me tango \frac{170}{9} mai i ngā taha e rua o te whārite.

$$y=\frac{2530}{9}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{2530}{9},p=\frac{850}{27}$$