Whakaoti i te {c}{x+y=21}{.25x+.05y=3.35}

$$\left. \begin{array} { c } { x + y = 21 } \\ { .25 x + .05 y = 3.35 } \end{array} \right.$$

Whakaoti mō x, y

$$x=11.5$$

$$y=9.5$$

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.

$$x+y=21,0.25x+0.05y=3.35$$

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

$$x+y=21$$

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

$$x=-y+21$$

Whakakapia te -y+21 mō te x ki tērā atu whārite, 0.25x+0.05y=3.35.

$$0.25\left(-y+21\right)+0.05y=3.35$$

Whakareatia 0.25 ki te -y+21.

$$-0.25y+5.25+0.05y=3.35$$

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

$$-0.2y+5.25=3.35$$

Me tango 5.25 mai i ngā taha e rua o te whārite.

$$-0.2y=-1.9$$

Me whakarea ngā taha e rua ki te -5.

$$y=9.5$$

Whakaurua te 9.5 mō y ki x=-y+21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-9.5+21$$

Tāpiri 21 ki te -9.5.

$$x=11.5$$

Kua oti te pūnaha te whakatau.

$$x=11.5,y=9.5$$

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.

$$x+y=21,0.25x+0.05y=3.35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.25&0.05\end{matrix}\right))\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.05}{0.05-0.25}&-\frac{1}{0.05-0.25}\\-\frac{0.25}{0.05-0.25}&\frac{1}{0.05-0.25}\end{matrix}\right)\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.25&5\\1.25&-5\end{matrix}\right)\left(\begin{matrix}21\\3.35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.25\times 21+5\times 3.35\\1.25\times 21-5\times 3.35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11.5\\9.5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=11.5,y=9.5$$

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.

$$x+y=21,0.25x+0.05y=3.35$$

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.

$$0.25x+0.25y=0.25\times 21,0.25x+0.05y=3.35$$

Whakarūnātia.

$$0.25x+0.25y=5.25,0.25x+0.05y=3.35$$

Me tango 0.25x+0.05y=3.35 mai i 0.25x+0.25y=5.25 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.25x-0.25x+0.25y-0.05y=5.25-3.35$$

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.

$$0.25y-0.05y=5.25-3.35$$

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

$$0.2y=5.25-3.35$$

Tāpiri 5.25 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.

$$0.2y=1.9$$

Me whakarea ngā taha e rua ki te 5.

$$y=9.5$$

Whakaurua te 9.5 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.

$$0.25x+0.05\times 9.5=3.35$$

Whakareatia 0.05 ki te 9.5 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.

$$0.25x+0.475=3.35$$

Me tango 0.475 mai i ngā taha e rua o te whārite.

$$0.25x=2.875$$

Me whakarea ngā taha e rua ki te 4.

$$x=11.5$$

Kua oti te pūnaha te whakatau.

$$x=11.5,y=9.5$$