Whakaoti i te {l}{x/4+y/3=7}{2/3x+y/2=14}

$$\left\{ \begin{array} { l } { \frac { x } { 4 } + \frac { y } { 3 } = 7 } \\ { \frac { 2 } { 3 } x + \frac { y } { 2 } = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=12$$

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.

$$\frac{1}{4}x+\frac{1}{3}y=7,\frac{2}{3}x+\frac{1}{2}y=14$$

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

$$\frac{1}{4}x+\frac{1}{3}y=7$$

Me tango \frac{y}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}x=-\frac{1}{3}y+7$$

Me whakarea ngā taha e rua ki te 4.

$$x=4\left(-\frac{1}{3}y+7\right)$$

Whakareatia 4 ki te -\frac{y}{3}+7.

$$x=-\frac{4}{3}y+28$$

Whakakapia te -\frac{4y}{3}+28 mō te x ki tērā atu whārite, \frac{2}{3}x+\frac{1}{2}y=14.

$$\frac{2}{3}\left(-\frac{4}{3}y+28\right)+\frac{1}{2}y=14$$

Whakareatia \frac{2}{3} ki te -\frac{4y}{3}+28.

$$-\frac{8}{9}y+\frac{56}{3}+\frac{1}{2}y=14$$

Tāpiri -\frac{8y}{9} ki te \frac{y}{2}.

$$-\frac{7}{18}y+\frac{56}{3}=14$$

Me tango \frac{56}{3} mai i ngā taha e rua o te whārite.

$$-\frac{7}{18}y=-\frac{14}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{18}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=12$$

Whakaurua te 12 mō y ki x=-\frac{4}{3}y+28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\times 12+28$$

Whakareatia -\frac{4}{3} ki te 12.

$$x=-16+28$$

Tāpiri 28 ki te -16.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=12$$

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.

$$\frac{1}{4}x+\frac{1}{3}y=7,\frac{2}{3}x+\frac{1}{2}y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}7\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}7\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{3}\\\frac{2}{3}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}7\\14\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{1}{4}\times \frac{1}{2}-\frac{1}{3}\times \frac{2}{3}}&-\frac{\frac{1}{3}}{\frac{1}{4}\times \frac{1}{2}-\frac{1}{3}\times \frac{2}{3}}\\-\frac{\frac{2}{3}}{\frac{1}{4}\times \frac{1}{2}-\frac{1}{3}\times \frac{2}{3}}&\frac{\frac{1}{4}}{\frac{1}{4}\times \frac{1}{2}-\frac{1}{3}\times \frac{2}{3}}\end{matrix}\right)\left(\begin{matrix}7\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{36}{7}&\frac{24}{7}\\\frac{48}{7}&-\frac{18}{7}\end{matrix}\right)\left(\begin{matrix}7\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{36}{7}\times 7+\frac{24}{7}\times 14\\\frac{48}{7}\times 7-\frac{18}{7}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=12$$

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.

$$\frac{1}{4}x+\frac{1}{3}y=7,\frac{2}{3}x+\frac{1}{2}y=14$$

Kia ōrite ai a \frac{x}{4} me \frac{2x}{3}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{2}{3} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{4}.

$$\frac{2}{3}\times \frac{1}{4}x+\frac{2}{3}\times \frac{1}{3}y=\frac{2}{3}\times 7,\frac{1}{4}\times \frac{2}{3}x+\frac{1}{4}\times \frac{1}{2}y=\frac{1}{4}\times 14$$

Whakarūnātia.

$$\frac{1}{6}x+\frac{2}{9}y=\frac{14}{3},\frac{1}{6}x+\frac{1}{8}y=\frac{7}{2}$$

Me tango \frac{1}{6}x+\frac{1}{8}y=\frac{7}{2} mai i \frac{1}{6}x+\frac{2}{9}y=\frac{14}{3} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{6}x-\frac{1}{6}x+\frac{2}{9}y-\frac{1}{8}y=\frac{14}{3}-\frac{7}{2}$$

Tāpiri \frac{x}{6} ki te -\frac{x}{6}. Ka whakakore atu ngā kupu \frac{x}{6} me -\frac{x}{6}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{2}{9}y-\frac{1}{8}y=\frac{14}{3}-\frac{7}{2}$$

Tāpiri \frac{2y}{9} ki te -\frac{y}{8}.

$$\frac{7}{72}y=\frac{14}{3}-\frac{7}{2}$$

Tāpiri \frac{14}{3} ki te -\frac{7}{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.

$$\frac{7}{72}y=\frac{7}{6}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{72}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=12$$

Whakaurua te 12 mō y ki \frac{2}{3}x+\frac{1}{2}y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{2}{3}x+\frac{1}{2}\times 12=14$$

Whakareatia \frac{1}{2} ki te 12.

$$\frac{2}{3}x+6=14$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$\frac{2}{3}x=8$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=12$$