Whakaoti i te {l}{x/3+y/2=8}{x/5+y/6=4}

$$\left. \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 2 } = 8 } \\ { \frac { x } { 5 } + \frac { y } { 6 } = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=15$$

$$y=6$$

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.

$$\frac{1}{3}x+\frac{1}{2}y=8,\frac{1}{5}x+\frac{1}{6}y=4$$

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

$$\frac{1}{3}x+\frac{1}{2}y=8$$

Me tango \frac{y}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x=-\frac{1}{2}y+8$$

Me whakarea ngā taha e rua ki te 3.

$$x=3\left(-\frac{1}{2}y+8\right)$$

Whakareatia 3 ki te -\frac{y}{2}+8.

$$x=-\frac{3}{2}y+24$$

Whakakapia te -\frac{3y}{2}+24 mō te x ki tērā atu whārite, \frac{1}{5}x+\frac{1}{6}y=4.

$$\frac{1}{5}\left(-\frac{3}{2}y+24\right)+\frac{1}{6}y=4$$

Whakareatia \frac{1}{5} ki te -\frac{3y}{2}+24.

$$-\frac{3}{10}y+\frac{24}{5}+\frac{1}{6}y=4$$

Tāpiri -\frac{3y}{10} ki te \frac{y}{6}.

$$-\frac{2}{15}y+\frac{24}{5}=4$$

Me tango \frac{24}{5} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{15}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{3}{2}y+24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 6+24$$

Whakareatia -\frac{3}{2} ki te 6.

$$x=-9+24$$

Tāpiri 24 ki te -9.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=6$$

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.

$$\frac{1}{3}x+\frac{1}{2}y=8,\frac{1}{5}x+\frac{1}{6}y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right))\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{3}&\frac{1}{2}\\\frac{1}{5}&\frac{1}{6}\end{matrix}\right))\left(\begin{matrix}8\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{6}}{\frac{1}{3}\times \frac{1}{6}-\frac{1}{2}\times \frac{1}{5}}&-\frac{\frac{1}{2}}{\frac{1}{3}\times \frac{1}{6}-\frac{1}{2}\times \frac{1}{5}}\\-\frac{\frac{1}{5}}{\frac{1}{3}\times \frac{1}{6}-\frac{1}{2}\times \frac{1}{5}}&\frac{\frac{1}{3}}{\frac{1}{3}\times \frac{1}{6}-\frac{1}{2}\times \frac{1}{5}}\end{matrix}\right)\left(\begin{matrix}8\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{4}&\frac{45}{4}\\\frac{9}{2}&-\frac{15}{2}\end{matrix}\right)\left(\begin{matrix}8\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{4}\times 8+\frac{45}{4}\times 4\\\frac{9}{2}\times 8-\frac{15}{2}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=15,y=6$$

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.

$$\frac{1}{3}x+\frac{1}{2}y=8,\frac{1}{5}x+\frac{1}{6}y=4$$

Kia ōrite ai a \frac{x}{3} me \frac{x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{3}.

$$\frac{1}{5}\times \frac{1}{3}x+\frac{1}{5}\times \frac{1}{2}y=\frac{1}{5}\times 8,\frac{1}{3}\times \frac{1}{5}x+\frac{1}{3}\times \frac{1}{6}y=\frac{1}{3}\times 4$$

Whakarūnātia.

$$\frac{1}{15}x+\frac{1}{10}y=\frac{8}{5},\frac{1}{15}x+\frac{1}{18}y=\frac{4}{3}$$

Me tango \frac{1}{15}x+\frac{1}{18}y=\frac{4}{3} mai i \frac{1}{15}x+\frac{1}{10}y=\frac{8}{5} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{15}x-\frac{1}{15}x+\frac{1}{10}y-\frac{1}{18}y=\frac{8}{5}-\frac{4}{3}$$

Tāpiri \frac{x}{15} ki te -\frac{x}{15}. Ka whakakore atu ngā kupu \frac{x}{15} me -\frac{x}{15}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{1}{10}y-\frac{1}{18}y=\frac{8}{5}-\frac{4}{3}$$

Tāpiri \frac{y}{10} ki te -\frac{y}{18}.

$$\frac{2}{45}y=\frac{8}{5}-\frac{4}{3}$$

Tāpiri \frac{8}{5} ki te -\frac{4}{3} 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.

$$\frac{2}{45}y=\frac{4}{15}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{45}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki \frac{1}{5}x+\frac{1}{6}y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{5}x+\frac{1}{6}\times 6=4$$

Whakareatia \frac{1}{6} ki te 6.

$$\frac{1}{5}x+1=4$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$\frac{1}{5}x=3$$

Me whakarea ngā taha e rua ki te 5.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=6$$