Whakaoti i te {l}{6x-1/3y=27}{4/5x+1/4y=25/4}

$$\left. \begin{array} { l } { 6 x - \frac { 1 } { 3 } y = 27 } \\ { \frac { 4 } { 5 } x + \frac { 1 } { 4 } y = \frac { 25 } { 4 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=9$$

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.

$$6x-\frac{1}{3}y=27,\frac{4}{5}x+\frac{1}{4}y=\frac{25}{4}$$

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

$$6x-\frac{1}{3}y=27$$

Me tāpiri \frac{y}{3} ki ngā taha e rua o te whārite.

$$6x=\frac{1}{3}y+27$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(\frac{1}{3}y+27\right)$$

Whakareatia \frac{1}{6} ki te \frac{y}{3}+27.

$$x=\frac{1}{18}y+\frac{9}{2}$$

Whakakapia te \frac{y}{18}+\frac{9}{2} mō te x ki tērā atu whārite, \frac{4}{5}x+\frac{1}{4}y=\frac{25}{4}.

$$\frac{4}{5}\left(\frac{1}{18}y+\frac{9}{2}\right)+\frac{1}{4}y=\frac{25}{4}$$

Whakareatia \frac{4}{5} ki te \frac{y}{18}+\frac{9}{2}.

$$\frac{2}{45}y+\frac{18}{5}+\frac{1}{4}y=\frac{25}{4}$$

Tāpiri \frac{2y}{45} ki te \frac{y}{4}.

$$\frac{53}{180}y+\frac{18}{5}=\frac{25}{4}$$

Me tango \frac{18}{5} mai i ngā taha e rua o te whārite.

$$\frac{53}{180}y=\frac{53}{20}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{53}{180}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=9$$

Whakaurua te 9 mō y ki x=\frac{1}{18}y+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{18}\times 9+\frac{9}{2}$$

Whakareatia \frac{1}{18} ki te 9.

$$x=\frac{1+9}{2}$$

Tāpiri \frac{9}{2} ki te \frac{1}{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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$

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.

$$6x-\frac{1}{3}y=27,\frac{4}{5}x+\frac{1}{4}y=\frac{25}{4}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-\frac{1}{3}\\\frac{4}{5}&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{4}}{6\times \frac{1}{4}-\left(-\frac{1}{3}\times \frac{4}{5}\right)}&-\frac{-\frac{1}{3}}{6\times \frac{1}{4}-\left(-\frac{1}{3}\times \frac{4}{5}\right)}\\-\frac{\frac{4}{5}}{6\times \frac{1}{4}-\left(-\frac{1}{3}\times \frac{4}{5}\right)}&\frac{6}{6\times \frac{1}{4}-\left(-\frac{1}{3}\times \frac{4}{5}\right)}\end{matrix}\right)\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{106}&\frac{10}{53}\\-\frac{24}{53}&\frac{180}{53}\end{matrix}\right)\left(\begin{matrix}27\\\frac{25}{4}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{106}\times 27+\frac{10}{53}\times \frac{25}{4}\\-\frac{24}{53}\times 27+\frac{180}{53}\times \frac{25}{4}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=9$$

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.

$$6x-\frac{1}{3}y=27,\frac{4}{5}x+\frac{1}{4}y=\frac{25}{4}$$

Kia ōrite ai a 6x me \frac{4x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{4}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$\frac{4}{5}\times 6x+\frac{4}{5}\left(-\frac{1}{3}\right)y=\frac{4}{5}\times 27,6\times \frac{4}{5}x+6\times \frac{1}{4}y=6\times \frac{25}{4}$$

Whakarūnātia.

$$\frac{24}{5}x-\frac{4}{15}y=\frac{108}{5},\frac{24}{5}x+\frac{3}{2}y=\frac{75}{2}$$

Me tango \frac{24}{5}x+\frac{3}{2}y=\frac{75}{2} mai i \frac{24}{5}x-\frac{4}{15}y=\frac{108}{5} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{24}{5}x-\frac{24}{5}x-\frac{4}{15}y-\frac{3}{2}y=\frac{108}{5}-\frac{75}{2}$$

Tāpiri \frac{24x}{5} ki te -\frac{24x}{5}. Ka whakakore atu ngā kupu \frac{24x}{5} me -\frac{24x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-\frac{4}{15}y-\frac{3}{2}y=\frac{108}{5}-\frac{75}{2}$$

Tāpiri -\frac{4y}{15} ki te -\frac{3y}{2}.

$$-\frac{53}{30}y=\frac{108}{5}-\frac{75}{2}$$

Tāpiri \frac{108}{5} ki te -\frac{75}{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.

$$-\frac{53}{30}y=-\frac{159}{10}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{53}{30}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=9$$

Whakaurua te 9 mō y ki \frac{4}{5}x+\frac{1}{4}y=\frac{25}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{4}{5}x+\frac{1}{4}\times 9=\frac{25}{4}$$

Whakareatia \frac{1}{4} ki te 9.

$$\frac{4}{5}x+\frac{9}{4}=\frac{25}{4}$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$