Whakaoti i te {l}{2x-2+4y=1/2}{8(y-1/2)=9(x+1)-4}

$$\left. \begin{array} { l } { 2 x - 2 + 4 y = \frac { 1 } { 2 } } \\ { 8 ( y - \frac { 1 } { 2 } ) = 9 ( x + 1 ) - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{4}{13}\approx -0.307692308$$

$$y=\frac{81}{104}\approx 0.778846154$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 2 ki ngā taha e rua.

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

Tāpirihia te \frac{1}{2} ki te 2, ka \frac{5}{2}.

$$2x+4y=\frac{5}{2}$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te y-\frac{1}{2}.

$$8y-4=9\left(x+1\right)-4$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te x+1.

$$8y-4=9x+9-4$$

Tangohia te 4 i te 9, ka 5.

$$8y-4=9x+5$$

Tangohia te 9x mai i ngā taha e rua.

$$8y-4-9x=5$$

Me tāpiri te 4 ki ngā taha e rua.

$$8y-9x=5+4$$

Tāpirihia te 5 ki te 4, ka 9.

$$8y-9x=9$$

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.

$$2x+4y=\frac{5}{2},-9x+8y=9$$

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

$$2x+4y=\frac{5}{2}$$

Me tango 4y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -4y+\frac{5}{2}.

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

Whakakapia te -2y+\frac{5}{4} mō te x ki tērā atu whārite, -9x+8y=9.

$$-9\left(-2y+\frac{5}{4}\right)+8y=9$$

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

$$18y-\frac{45}{4}+8y=9$$

Tāpiri 18y ki te 8y.

$$26y-\frac{45}{4}=9$$

Me tāpiri \frac{45}{4} ki ngā taha e rua o te whārite.

$$26y=\frac{81}{4}$$

Whakawehea ngā taha e rua ki te 26.

$$y=\frac{81}{104}$$

Whakaurua te \frac{81}{104} mō y ki x=-2y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{81}{104}+\frac{5}{4}$$

Whakareatia -2 ki te \frac{81}{104}.

$$x=-\frac{81}{52}+\frac{5}{4}$$

Tāpiri \frac{5}{4} ki te -\frac{81}{52} 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.

$$x=-\frac{4}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{13},y=\frac{81}{104}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 2 ki ngā taha e rua.

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

Tāpirihia te \frac{1}{2} ki te 2, ka \frac{5}{2}.

$$2x+4y=\frac{5}{2}$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te y-\frac{1}{2}.

$$8y-4=9\left(x+1\right)-4$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te x+1.

$$8y-4=9x+9-4$$

Tangohia te 4 i te 9, ka 5.

$$8y-4=9x+5$$

Tangohia te 9x mai i ngā taha e rua.

$$8y-4-9x=5$$

Me tāpiri te 4 ki ngā taha e rua.

$$8y-9x=5+4$$

Tāpirihia te 5 ki te 4, ka 9.

$$8y-9x=9$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x+4y=\frac{5}{2},-9x+8y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&4\\-9&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&4\\-9&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&4\\-9&8\end{matrix}\right))\left(\begin{matrix}2&4\\-9&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-9&8\end{matrix}\right))\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-9&8\end{matrix}\right))\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\-9&8\end{matrix}\right))\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{2\times 8-4\left(-9\right)}&-\frac{4}{2\times 8-4\left(-9\right)}\\-\frac{-9}{2\times 8-4\left(-9\right)}&\frac{2}{2\times 8-4\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&-\frac{1}{13}\\\frac{9}{52}&\frac{1}{26}\end{matrix}\right)\left(\begin{matrix}\frac{5}{2}\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times \frac{5}{2}-\frac{1}{13}\times 9\\\frac{9}{52}\times \frac{5}{2}+\frac{1}{26}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{13}\\\frac{81}{104}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{4}{13},y=\frac{81}{104}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 2 ki ngā taha e rua.

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

Tāpirihia te \frac{1}{2} ki te 2, ka \frac{5}{2}.

$$2x+4y=\frac{5}{2}$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te y-\frac{1}{2}.

$$8y-4=9\left(x+1\right)-4$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te x+1.

$$8y-4=9x+9-4$$

Tangohia te 4 i te 9, ka 5.

$$8y-4=9x+5$$

Tangohia te 9x mai i ngā taha e rua.

$$8y-4-9x=5$$

Me tāpiri te 4 ki ngā taha e rua.

$$8y-9x=5+4$$

Tāpirihia te 5 ki te 4, ka 9.

$$8y-9x=9$$

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.

$$2x+4y=\frac{5}{2},-9x+8y=9$$

Kia ōrite ai a 2x me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-9\times 2x-9\times 4y=-9\times \frac{5}{2},2\left(-9\right)x+2\times 8y=2\times 9$$

Whakarūnātia.

$$-18x-36y=-\frac{45}{2},-18x+16y=18$$

Me tango -18x+16y=18 mai i -18x-36y=-\frac{45}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-18x+18x-36y-16y=-\frac{45}{2}-18$$

Tāpiri -18x ki te 18x. Ka whakakore atu ngā kupu -18x me 18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-36y-16y=-\frac{45}{2}-18$$

Tāpiri -36y ki te -16y.

$$-52y=-\frac{45}{2}-18$$

Tāpiri -\frac{45}{2} ki te -18.

$$-52y=-\frac{81}{2}$$

Whakawehea ngā taha e rua ki te -52.

$$y=\frac{81}{104}$$

Whakaurua te \frac{81}{104} mō y ki -9x+8y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x+8\times \frac{81}{104}=9$$

Whakareatia 8 ki te \frac{81}{104}.

$$-9x+\frac{81}{13}=9$$

Me tango \frac{81}{13} mai i ngā taha e rua o te whārite.

$$-9x=\frac{36}{13}$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{4}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{4}{13},y=\frac{81}{104}$$