Whakaoti i te {c}{2.7x+3.1y=42.5}{x+y=15}

$$\left. \begin{array} { c } { 2.7 x + 3.1 y = 42.5 } \\ { x + y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=5$$

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.

$$2.7x+3.1y=42.5,x+y=15$$

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

$$2.7x+3.1y=42.5$$

Me tango \frac{31y}{10} mai i ngā taha e rua o te whārite.

$$2.7x=-3.1y+42.5$$

Whakawehea ngā taha e rua o te whārite ki te 2.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{10}{27}\left(-3.1y+42.5\right)$$

Whakareatia \frac{10}{27} ki te -\frac{31y}{10}+42.5.

$$x=-\frac{31}{27}y+\frac{425}{27}$$

Whakakapia te \frac{-31y+425}{27} mō te x ki tērā atu whārite, x+y=15.

$$-\frac{31}{27}y+\frac{425}{27}+y=15$$

Tāpiri -\frac{31y}{27} ki te y.

$$-\frac{4}{27}y+\frac{425}{27}=15$$

Me tango \frac{425}{27} mai i ngā taha e rua o te whārite.

$$-\frac{4}{27}y=-\frac{20}{27}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{27}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{31}{27}y+\frac{425}{27}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{31}{27}\times 5+\frac{425}{27}$$

Whakareatia -\frac{31}{27} ki te 5.

$$x=\frac{-155+425}{27}$$

Tāpiri \frac{425}{27} ki te -\frac{155}{27} 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.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=5$$

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.

$$2.7x+3.1y=42.5,x+y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right))\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right))\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right))\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.7&3.1\\1&1\end{matrix}\right))\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2.7-3.1}&-\frac{3.1}{2.7-3.1}\\-\frac{1}{2.7-3.1}&\frac{2.7}{2.7-3.1}\end{matrix}\right)\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2.5&7.75\\2.5&-6.75\end{matrix}\right)\left(\begin{matrix}42.5\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2.5\times 42.5+7.75\times 15\\2.5\times 42.5-6.75\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=5$$

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.

$$2.7x+3.1y=42.5,x+y=15$$

Kia ōrite ai a \frac{27x}{10} me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.7.

$$2.7x+3.1y=42.5,2.7x+2.7y=2.7\times 15$$

Whakarūnātia.

$$2.7x+3.1y=42.5,2.7x+2.7y=40.5$$

Me tango 2.7x+2.7y=40.5 mai i 2.7x+3.1y=42.5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2.7x-2.7x+3.1y-2.7y=\frac{85-81}{2}$$

Tāpiri \frac{27x}{10} ki te -\frac{27x}{10}. Ka whakakore atu ngā kupu \frac{27x}{10} me -\frac{27x}{10}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3.1y-2.7y=\frac{85-81}{2}$$

Tāpiri \frac{31y}{10} ki te -\frac{27y}{10}.

$$0.4y=\frac{85-81}{2}$$

Tāpiri 42.5 ki te -40.5 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.

$$0.4y=2$$

Whakawehea ngā taha e rua o te whārite ki te 0.4, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x+y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5=15$$

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

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=5$$