Whakaoti i te {c}{x+y=27}{.25x+.05y=3.35}

$$\left. \begin{array} { c } { x + y = 27 } \\ { .25 x + .05 y = 3.35 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=17$$

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.

$$x+y=27,0.25x+0.05y=3.35$$

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

$$x+y=27$$

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

$$x=-y+27$$

Whakakapia te -y+27 mō te x ki tērā atu whārite, 0.25x+0.05y=3.35.

$$0.25\left(-y+27\right)+0.05y=3.35$$

Whakareatia 0.25 ki te -y+27.

$$-0.25y+6.75+0.05y=3.35$$

Tāpiri -\frac{y}{4} ki te \frac{y}{20}.

$$-0.2y+6.75=3.35$$

Me tango 6.75 mai i ngā taha e rua o te whārite.

$$-0.2y=-3.4$$

Me whakarea ngā taha e rua ki te -5.

$$y=17$$

Whakaurua te 17 mō y ki x=-y+27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-17+27$$

Tāpiri 27 ki te -17.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=17$$

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.

$$x+y=27,0.25x+0.05y=3.35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.05}{0.05-0.25}&-\frac{1}{0.05-0.25}\\-\frac{0.25}{0.05-0.25}&\frac{1}{0.05-0.25}\end{matrix}\right)\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.25&5\\1.25&-5\end{matrix}\right)\left(\begin{matrix}27\\3.35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.25\times 27+5\times 3.35\\1.25\times 27-5\times 3.35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\17\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=17$$

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.

$$x+y=27,0.25x+0.05y=3.35$$

Kia ōrite ai a x me \frac{x}{4}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.25 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$0.25x+0.25y=0.25\times 27,0.25x+0.05y=3.35$$

Whakarūnātia.

$$0.25x+0.25y=6.75,0.25x+0.05y=3.35$$

Me tango 0.25x+0.05y=3.35 mai i 0.25x+0.25y=6.75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.25x-0.25x+0.25y-0.05y=6.75-3.35$$

Tāpiri \frac{x}{4} ki te -\frac{x}{4}. Ka whakakore atu ngā kupu \frac{x}{4} me -\frac{x}{4}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.25y-0.05y=6.75-3.35$$

Tāpiri \frac{y}{4} ki te -\frac{y}{20}.

$$0.2y=6.75-3.35$$

Tāpiri 6.75 ki te -3.35 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.

$$0.2y=3.4$$

Me whakarea ngā taha e rua ki te 5.

$$y=17$$

Whakaurua te 17 mō y ki 0.25x+0.05y=3.35. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.25x+0.05\times 17=3.35$$

Whakareatia 0.05 ki te 17.

$$0.25x+0.85=3.35$$

Me tango 0.85 mai i ngā taha e rua o te whārite.

$$0.25x=2.5$$

Me whakarea ngā taha e rua ki te 4.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=17$$