Whakaoti i te {l}{6x+3y=25.95}{4x+6y=26.70}

$$\left. \begin{array} { l } { 6 x + 3 y = 25.95 } \\ { 4 x + 6 y = 26.70 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3.15$$

$$y=2.35$$

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.

$$6x+3y=25.95,4x+6y=26.7$$

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

$$6x+3y=25.95$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$6x=-3y+25.95$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-3y+25.95\right)$$

Whakareatia \frac{1}{6} ki te -3y+25.95.

$$x=-\frac{1}{2}y+\frac{173}{40}$$

Whakakapia te -\frac{y}{2}+\frac{173}{40} mō te x ki tērā atu whārite, 4x+6y=26.7.

$$4\left(-\frac{1}{2}y+\frac{173}{40}\right)+6y=26.7$$

Whakareatia 4 ki te -\frac{y}{2}+\frac{173}{40}.

$$-2y+\frac{173}{10}+6y=26.7$$

Tāpiri -2y ki te 6y.

$$4y+\frac{173}{10}=26.7$$

Me tango \frac{173}{10} mai i ngā taha e rua o te whārite.

$$4y=\frac{47}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{47}{20}$$

Whakaurua te \frac{47}{20} mō y ki x=-\frac{1}{2}y+\frac{173}{40}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{47}{20}+\frac{173}{40}$$

Whakareatia -\frac{1}{2} ki te \frac{47}{20} 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.

$$x=\frac{-47+173}{40}$$

Tāpiri \frac{173}{40} ki te -\frac{47}{40} 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.

$$x=\frac{63}{20}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{63}{20},y=\frac{47}{20}$$

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.

$$6x+3y=25.95,4x+6y=26.7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&3\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&3\\4&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&3\\4&6\end{matrix}\right))\left(\begin{matrix}6&3\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\4&6\end{matrix}\right))\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\4&6\end{matrix}\right))\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\4&6\end{matrix}\right))\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{6\times 6-3\times 4}&-\frac{3}{6\times 6-3\times 4}\\-\frac{4}{6\times 6-3\times 4}&\frac{6}{6\times 6-3\times 4}\end{matrix}\right)\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{8}\\-\frac{1}{6}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}25.95\\26.7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 25.95-\frac{1}{8}\times 26.7\\-\frac{1}{6}\times 25.95+\frac{1}{4}\times 26.7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{63}{20}\\\frac{47}{20}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{63}{20},y=\frac{47}{20}$$

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.

$$6x+3y=25.95,4x+6y=26.7$$

Kia ōrite ai a 6x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$4\times 6x+4\times 3y=4\times 25.95,6\times 4x+6\times 6y=6\times 26.7$$

Whakarūnātia.

$$24x+12y=103.8,24x+36y=160.2$$

Me tango 24x+36y=160.2 mai i 24x+12y=103.8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x+12y-36y=\frac{519-801}{5}$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-36y=\frac{519-801}{5}$$

Tāpiri 12y ki te -36y.

$$-24y=\frac{519-801}{5}$$

Tāpiri 103.8 ki te -160.2 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.

$$-24y=-56.4$$

Whakawehea ngā taha e rua ki te -24.

$$y=\frac{47}{20}$$

Whakaurua te \frac{47}{20} mō y ki 4x+6y=26.7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+6\times \frac{47}{20}=26.7$$

Whakareatia 6 ki te \frac{47}{20}.

$$4x+\frac{141}{10}=26.7$$

Me tango \frac{141}{10} mai i ngā taha e rua o te whārite.

$$4x=\frac{63}{5}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{63}{20}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{63}{20},y=\frac{47}{20}$$