Whakaoti i te {l}{4x+8y=64}{2x-8y=86}

$$\left. \begin{array} { l } { 4 x + 8 y = 64 } \\ { 2 x - 8 y = 86 } \end{array} \right.$$

Whakaoti mō x, y

$$x=25$$

$$y = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

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.

$$4x+8y=64,2x-8y=86$$

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

$$4x+8y=64$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$4x=-8y+64$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-8y+64\right)$$

Whakareatia \frac{1}{4} ki te -8y+64.

$$x=-2y+16$$

Whakakapia te -2y+16 mō te x ki tērā atu whārite, 2x-8y=86.

$$2\left(-2y+16\right)-8y=86$$

Whakareatia 2 ki te -2y+16.

$$-4y+32-8y=86$$

Tāpiri -4y ki te -8y.

$$-12y+32=86$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$-12y=54$$

Whakawehea ngā taha e rua ki te -12.

$$y=-\frac{9}{2}$$

Whakaurua te -\frac{9}{2} mō y ki x=-2y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-\frac{9}{2}\right)+16$$

Whakareatia -2 ki te -\frac{9}{2}.

$$x=9+16$$

Tāpiri 16 ki te 9.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=-\frac{9}{2}$$

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.

$$4x+8y=64,2x-8y=86$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&8\\2&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}64\\86\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&8\\2&-8\end{matrix}\right))\left(\begin{matrix}4&8\\2&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&8\\2&-8\end{matrix}\right))\left(\begin{matrix}64\\86\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&8\\2&-8\end{matrix}\right))\left(\begin{matrix}64\\86\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&8\\2&-8\end{matrix}\right))\left(\begin{matrix}64\\86\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{4\left(-8\right)-8\times 2}&-\frac{8}{4\left(-8\right)-8\times 2}\\-\frac{2}{4\left(-8\right)-8\times 2}&\frac{4}{4\left(-8\right)-8\times 2}\end{matrix}\right)\left(\begin{matrix}64\\86\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{1}{24}&-\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}64\\86\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 64+\frac{1}{6}\times 86\\\frac{1}{24}\times 64-\frac{1}{12}\times 86\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\-\frac{9}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=25,y=-\frac{9}{2}$$

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.

$$4x+8y=64,2x-8y=86$$

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

$$2\times 4x+2\times 8y=2\times 64,4\times 2x+4\left(-8\right)y=4\times 86$$

Whakarūnātia.

$$8x+16y=128,8x-32y=344$$

Me tango 8x-32y=344 mai i 8x+16y=128 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+16y+32y=128-344$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16y+32y=128-344$$

Tāpiri 16y ki te 32y.

$$48y=128-344$$

Tāpiri 128 ki te -344.

$$48y=-216$$

Whakawehea ngā taha e rua ki te 48.

$$y=-\frac{9}{2}$$

Whakaurua te -\frac{9}{2} mō y ki 2x-8y=86. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-8\left(-\frac{9}{2}\right)=86$$

Whakareatia -8 ki te -\frac{9}{2}.

$$2x+36=86$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$2x=50$$

Whakawehea ngā taha e rua ki te 2.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=-\frac{9}{2}$$