Whakaoti i te {l}{y+25x=45}{y+.30x=35}

$$\left. \begin{array} { l } { y + 25 x = 45 } \\ { y + .30 x = 35 } \end{array} \right.$$

Whakaoti mō y, x

$$x=\frac{100}{247}\approx 0.4048583$$

$$y = \frac{8615}{247} = 34\frac{217}{247} \approx 34.87854251$$

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.

$$y+25x=45,y+0.3x=35$$

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

$$y+25x=45$$

Me tango 25x mai i ngā taha e rua o te whārite.

$$y=-25x+45$$

Whakakapia te -25x+45 mō te y ki tērā atu whārite, y+0.3x=35.

$$-25x+45+0.3x=35$$

Tāpiri -25x ki te \frac{3x}{10}.

$$-24.7x+45=35$$

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

$$-24.7x=-10$$

Whakawehea ngā taha e rua o te whārite ki te -24.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{100}{247}$$

Whakaurua te \frac{100}{247} mō x ki y=-25x+45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-25\times \frac{100}{247}+45$$

Whakareatia -25 ki te \frac{100}{247}.

$$y=-\frac{2500}{247}+45$$

Tāpiri 45 ki te -\frac{2500}{247}.

$$y=\frac{8615}{247}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{8615}{247},x=\frac{100}{247}$$

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.

$$y+25x=45,y+0.3x=35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}45\\35\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&25\\1&0.3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right))\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right))\left(\begin{matrix}45\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right))\left(\begin{matrix}45\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&25\\1&0.3\end{matrix}\right))\left(\begin{matrix}45\\35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{0.3}{0.3-25}&-\frac{25}{0.3-25}\\-\frac{1}{0.3-25}&\frac{1}{0.3-25}\end{matrix}\right)\left(\begin{matrix}45\\35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{247}&\frac{250}{247}\\\frac{10}{247}&-\frac{10}{247}\end{matrix}\right)\left(\begin{matrix}45\\35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{247}\times 45+\frac{250}{247}\times 35\\\frac{10}{247}\times 45-\frac{10}{247}\times 35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8615}{247}\\\frac{100}{247}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{8615}{247},x=\frac{100}{247}$$

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.

$$y+25x=45,y+0.3x=35$$

Me tango y+0.3x=35 mai i y+25x=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+25x-0.3x=45-35$$

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.

$$25x-0.3x=45-35$$

Tāpiri 25x ki te -\frac{3x}{10}.

$$24.7x=45-35$$

Tāpiri 45 ki te -35.

$$24.7x=10$$

Whakawehea ngā taha e rua o te whārite ki te 24.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{100}{247}$$

Whakaurua te \frac{100}{247} mō x ki y+0.3x=35. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+0.3\times \frac{100}{247}=35$$

Whakareatia 0.3 ki te \frac{100}{247} 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.

$$y+\frac{30}{247}=35$$

Me tango \frac{30}{247} mai i ngā taha e rua o te whārite.

$$y=\frac{8615}{247}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{8615}{247},x=\frac{100}{247}$$