Whakaoti i te {l}{6x+8y=20}{5y+3x=8}

$$\left\{ \begin{array} { l } { 6 x + 8 y = 20 } \\ { 5 y + 3 x = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-2$$

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.

$$6x+8y=20,3x+5y=8$$

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

$$6x+8y=20$$

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

$$6x=-8y+20$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-8y+20\right)$$

Whakareatia \frac{1}{6} ki te -8y+20.

$$x=-\frac{4}{3}y+\frac{10}{3}$$

Whakakapia te \frac{-4y+10}{3} mō te x ki tērā atu whārite, 3x+5y=8.

$$3\left(-\frac{4}{3}y+\frac{10}{3}\right)+5y=8$$

Whakareatia 3 ki te \frac{-4y+10}{3}.

$$-4y+10+5y=8$$

Tāpiri -4y ki te 5y.

$$y+10=8$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{4}{3}y+\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-2\right)+\frac{10}{3}$$

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

$$x=\frac{8+10}{3}$$

Tāpiri \frac{10}{3} ki te \frac{8}{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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-2$$

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.

$$6x+8y=20,3x+5y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&8\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&8\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&8\\3&5\end{matrix}\right))\left(\begin{matrix}6&8\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&8\\3&5\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&8\\3&5\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&8\\3&5\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6\times 5-8\times 3}&-\frac{8}{6\times 5-8\times 3}\\-\frac{3}{6\times 5-8\times 3}&\frac{6}{6\times 5-8\times 3}\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}&-\frac{4}{3}\\-\frac{1}{2}&1\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\times 20-\frac{4}{3}\times 8\\-\frac{1}{2}\times 20+8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-2$$

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.

$$6x+8y=20,3x+5y=8$$

Kia ōrite ai a 6x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$3\times 6x+3\times 8y=3\times 20,6\times 3x+6\times 5y=6\times 8$$

Whakarūnātia.

$$18x+24y=60,18x+30y=48$$

Me tango 18x+30y=48 mai i 18x+24y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+24y-30y=60-48$$

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.

$$24y-30y=60-48$$

Tāpiri 24y ki te -30y.

$$-6y=60-48$$

Tāpiri 60 ki te -48.

$$-6y=12$$

Whakawehea ngā taha e rua ki te -6.

$$y=-2$$

Whakaurua te -2 mō y ki 3x+5y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\left(-2\right)=8$$

Whakareatia 5 ki te -2.

$$3x-10=8$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$3x=18$$

Whakawehea ngā taha e rua ki te 3.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-2$$