Whakaoti i te {l}{3/2x+6y=19/8}{1/2x-9y=-23/8}

$$\left. \begin{array} { l } { 3 / 2 x + 6 y = \frac { 19 } { 8 } } \\ { 1 / 2 x - 9 y = \frac { - 23 } { 8 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{4}=0.25$$

$$y=\frac{1}{3}\approx 0.333333333$$

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.

$$\frac{3}{2}x+6y=\frac{19}{8},\frac{1}{2}x-9y=-\frac{23}{8}$$

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

$$\frac{3}{2}x+6y=\frac{19}{8}$$

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

$$\frac{3}{2}x=-6y+\frac{19}{8}$$

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

$$x=\frac{2}{3}\left(-6y+\frac{19}{8}\right)$$

Whakareatia \frac{2}{3} ki te -6y+\frac{19}{8}.

$$x=-4y+\frac{19}{12}$$

Whakakapia te -4y+\frac{19}{12} mō te x ki tērā atu whārite, \frac{1}{2}x-9y=-\frac{23}{8}.

$$\frac{1}{2}\left(-4y+\frac{19}{12}\right)-9y=-\frac{23}{8}$$

Whakareatia \frac{1}{2} ki te -4y+\frac{19}{12}.

$$-2y+\frac{19}{24}-9y=-\frac{23}{8}$$

Tāpiri -2y ki te -9y.

$$-11y+\frac{19}{24}=-\frac{23}{8}$$

Me tango \frac{19}{24} mai i ngā taha e rua o te whārite.

$$-11y=-\frac{11}{3}$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{1}{3}$$

Whakaurua te \frac{1}{3} mō y ki x=-4y+\frac{19}{12}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times \frac{1}{3}+\frac{19}{12}$$

Whakareatia -4 ki te \frac{1}{3}.

$$x=-\frac{4}{3}+\frac{19}{12}$$

Tāpiri \frac{19}{12} 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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{1}{3}$$

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.

$$\frac{3}{2}x+6y=\frac{19}{8},\frac{1}{2}x-9y=-\frac{23}{8}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right))\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right))\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right))\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&6\\\frac{1}{2}&-9\end{matrix}\right))\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{\frac{3}{2}\left(-9\right)-6\times \frac{1}{2}}&-\frac{6}{\frac{3}{2}\left(-9\right)-6\times \frac{1}{2}}\\-\frac{\frac{1}{2}}{\frac{3}{2}\left(-9\right)-6\times \frac{1}{2}}&\frac{\frac{3}{2}}{\frac{3}{2}\left(-9\right)-6\times \frac{1}{2}}\end{matrix}\right)\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{11}&\frac{4}{11}\\\frac{1}{33}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}\frac{19}{8}\\-\frac{23}{8}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{11}\times \frac{19}{8}+\frac{4}{11}\left(-\frac{23}{8}\right)\\\frac{1}{33}\times \frac{19}{8}-\frac{1}{11}\left(-\frac{23}{8}\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{4},y=\frac{1}{3}$$

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.

$$\frac{3}{2}x+6y=\frac{19}{8},\frac{1}{2}x-9y=-\frac{23}{8}$$

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

$$\frac{1}{2}\times \frac{3}{2}x+\frac{1}{2}\times 6y=\frac{1}{2}\times \frac{19}{8},\frac{3}{2}\times \frac{1}{2}x+\frac{3}{2}\left(-9\right)y=\frac{3}{2}\left(-\frac{23}{8}\right)$$

Whakarūnātia.

$$\frac{3}{4}x+3y=\frac{19}{16},\frac{3}{4}x-\frac{27}{2}y=-\frac{69}{16}$$

Me tango \frac{3}{4}x-\frac{27}{2}y=-\frac{69}{16} mai i \frac{3}{4}x+3y=\frac{19}{16} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{4}x-\frac{3}{4}x+3y+\frac{27}{2}y=\frac{19+69}{16}$$

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.

$$3y+\frac{27}{2}y=\frac{19+69}{16}$$

Tāpiri 3y ki te \frac{27y}{2}.

$$\frac{33}{2}y=\frac{19+69}{16}$$

Tāpiri \frac{19}{16} ki te \frac{69}{16} 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.

$$\frac{33}{2}y=\frac{11}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{33}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1}{3}$$

Whakaurua te \frac{1}{3} mō y ki \frac{1}{2}x-9y=-\frac{23}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{2}x-9\times \frac{1}{3}=-\frac{23}{8}$$

Whakareatia -9 ki te \frac{1}{3}.

$$\frac{1}{2}x-3=-\frac{23}{8}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$\frac{1}{2}x=\frac{1}{8}$$

Me whakarea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{1}{3}$$