Whakaoti i te {l}{3/4x+1/4y=13/2}{x-3/4y=frac{13}{2}}

$$\left. \begin{array} { l } { \frac { 3 } { 4 } x + \frac { 1 } { 4 } y = \frac { 13 } { 2 } } \\ { x - \frac { 3 } { 4 } y = \frac { 13 } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=2$$

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.

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2},x-\frac{3}{4}y=\frac{13}{2}$$

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

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2}$$

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

$$\frac{3}{4}x=-\frac{1}{4}y+\frac{13}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{4}{3}\left(-\frac{1}{4}y+\frac{13}{2}\right)$$

Whakareatia \frac{4}{3} ki te -\frac{y}{4}+\frac{13}{2}.

$$x=-\frac{1}{3}y+\frac{26}{3}$$

Whakakapia te \frac{-y+26}{3} mō te x ki tērā atu whārite, x-\frac{3}{4}y=\frac{13}{2}.

$$-\frac{1}{3}y+\frac{26}{3}-\frac{3}{4}y=\frac{13}{2}$$

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

$$-\frac{13}{12}y+\frac{26}{3}=\frac{13}{2}$$

Me tango \frac{26}{3} mai i ngā taha e rua o te whārite.

$$-\frac{13}{12}y=-\frac{13}{6}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{12}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{1}{3}y+\frac{26}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 2+\frac{26}{3}$$

Whakareatia -\frac{1}{3} ki te 2.

$$x=\frac{-2+26}{3}$$

Tāpiri \frac{26}{3} ki te -\frac{2}{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.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=2$$

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.

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2},x-\frac{3}{4}y=\frac{13}{2}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\1&-\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{3}{4}}{\frac{3}{4}\left(-\frac{3}{4}\right)-\frac{1}{4}}&-\frac{\frac{1}{4}}{\frac{3}{4}\left(-\frac{3}{4}\right)-\frac{1}{4}}\\-\frac{1}{\frac{3}{4}\left(-\frac{3}{4}\right)-\frac{1}{4}}&\frac{\frac{3}{4}}{\frac{3}{4}\left(-\frac{3}{4}\right)-\frac{1}{4}}\end{matrix}\right)\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{13}&\frac{4}{13}\\\frac{16}{13}&-\frac{12}{13}\end{matrix}\right)\left(\begin{matrix}\frac{13}{2}\\\frac{13}{2}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{13}\times \frac{13}{2}+\frac{4}{13}\times \frac{13}{2}\\\frac{16}{13}\times \frac{13}{2}-\frac{12}{13}\times \frac{13}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=2$$

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.

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2},x-\frac{3}{4}y=\frac{13}{2}$$

Kia ōrite ai a \frac{3x}{4} 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 \frac{3}{4}.

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2},\frac{3}{4}x+\frac{3}{4}\left(-\frac{3}{4}\right)y=\frac{3}{4}\times \frac{13}{2}$$

Whakarūnātia.

$$\frac{3}{4}x+\frac{1}{4}y=\frac{13}{2},\frac{3}{4}x-\frac{9}{16}y=\frac{39}{8}$$

Me tango \frac{3}{4}x-\frac{9}{16}y=\frac{39}{8} mai i \frac{3}{4}x+\frac{1}{4}y=\frac{13}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{4}x-\frac{3}{4}x+\frac{1}{4}y+\frac{9}{16}y=\frac{13}{2}-\frac{39}{8}$$

Tāpiri \frac{3x}{4} ki te -\frac{3x}{4}. Ka whakakore atu ngā kupu \frac{3x}{4} me -\frac{3x}{4}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{1}{4}y+\frac{9}{16}y=\frac{13}{2}-\frac{39}{8}$$

Tāpiri \frac{y}{4} ki te \frac{9y}{16}.

$$\frac{13}{16}y=\frac{13}{2}-\frac{39}{8}$$

Tāpiri \frac{13}{2} ki te -\frac{39}{8} 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.

$$\frac{13}{16}y=\frac{13}{8}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{16}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x-\frac{3}{4}y=\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{3}{4}\times 2=\frac{13}{2}$$

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

$$x-\frac{3}{2}=\frac{13}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=2$$