Whakaoti i te {l}{2x+16y=22}{4x+8y=20}

$$\left. \begin{array} { l } { 2 x + 16 y = 22 } \\ { 4 x + 8 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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.

$$2x+16y=22,4x+8y=20$$

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

$$2x+16y=22$$

Me tango 16y mai i ngā taha e rua o te whārite.

$$2x=-16y+22$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-16y+22\right)$$

Whakareatia \frac{1}{2} ki te -16y+22.

$$x=-8y+11$$

Whakakapia te -8y+11 mō te x ki tērā atu whārite, 4x+8y=20.

$$4\left(-8y+11\right)+8y=20$$

Whakareatia 4 ki te -8y+11.

$$-32y+44+8y=20$$

Tāpiri -32y ki te 8y.

$$-24y+44=20$$

Me tango 44 mai i ngā taha e rua o te whārite.

$$-24y=-24$$

Whakawehea ngā taha e rua ki te -24.

$$y=1$$

Whakaurua te 1 mō y ki x=-8y+11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-8+11$$

Tāpiri 11 ki te -8.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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.

$$2x+16y=22,4x+8y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&16\\4&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}22\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&16\\4&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&16\\4&8\end{matrix}\right))\left(\begin{matrix}2&16\\4&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&16\\4&8\end{matrix}\right))\left(\begin{matrix}22\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&16\\4&8\end{matrix}\right))\left(\begin{matrix}22\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&16\\4&8\end{matrix}\right))\left(\begin{matrix}22\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{2\times 8-16\times 4}&-\frac{16}{2\times 8-16\times 4}\\-\frac{4}{2\times 8-16\times 4}&\frac{2}{2\times 8-16\times 4}\end{matrix}\right)\left(\begin{matrix}22\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{3}\\\frac{1}{12}&-\frac{1}{24}\end{matrix}\right)\left(\begin{matrix}22\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 22+\frac{1}{3}\times 20\\\frac{1}{12}\times 22-\frac{1}{24}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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.

$$2x+16y=22,4x+8y=20$$

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

$$4\times 2x+4\times 16y=4\times 22,2\times 4x+2\times 8y=2\times 20$$

Whakarūnātia.

$$8x+64y=88,8x+16y=40$$

Me tango 8x+16y=40 mai i 8x+64y=88 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+64y-16y=88-40$$

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.

$$64y-16y=88-40$$

Tāpiri 64y ki te -16y.

$$48y=88-40$$

Tāpiri 88 ki te -40.

$$48y=48$$

Whakawehea ngā taha e rua ki te 48.

$$y=1$$

Whakaurua te 1 mō y ki 4x+8y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+8=20$$

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

$$4x=12$$

Whakawehea ngā taha e rua ki te 4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$