Whakaoti i te {l}{5x+2y=26}{3x+8y=53}

$$\left. \begin{array} { l } { 5 x + 2 y = 26 } \\ { 3 x + 8 y = 53 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

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.

$$5x+2y=26,3x+8y=53$$

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

$$5x+2y=26$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+26$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+26\right)$$

Whakareatia \frac{1}{5} ki te -2y+26.

$$x=-\frac{2}{5}y+\frac{26}{5}$$

Whakakapia te \frac{-2y+26}{5} mō te x ki tērā atu whārite, 3x+8y=53.

$$3\left(-\frac{2}{5}y+\frac{26}{5}\right)+8y=53$$

Whakareatia 3 ki te \frac{-2y+26}{5}.

$$-\frac{6}{5}y+\frac{78}{5}+8y=53$$

Tāpiri -\frac{6y}{5} ki te 8y.

$$\frac{34}{5}y+\frac{78}{5}=53$$

Me tango \frac{78}{5} mai i ngā taha e rua o te whārite.

$$\frac{34}{5}y=\frac{187}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{34}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te \frac{11}{2} mō y ki x=-\frac{2}{5}y+\frac{26}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\times \frac{11}{2}+\frac{26}{5}$$

Whakareatia -\frac{2}{5} ki te \frac{11}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-11+26}{5}$$

Tāpiri \frac{26}{5} ki te -\frac{11}{5} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{11}{2}$$

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.

$$5x+2y=26,3x+8y=53$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\3&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}26\\53\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\3&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\3&8\end{matrix}\right))\left(\begin{matrix}5&2\\3&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&8\end{matrix}\right))\left(\begin{matrix}26\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&8\end{matrix}\right))\left(\begin{matrix}26\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&8\end{matrix}\right))\left(\begin{matrix}26\\53\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5\times 8-2\times 3}&-\frac{2}{5\times 8-2\times 3}\\-\frac{3}{5\times 8-2\times 3}&\frac{5}{5\times 8-2\times 3}\end{matrix}\right)\left(\begin{matrix}26\\53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{17}&-\frac{1}{17}\\-\frac{3}{34}&\frac{5}{34}\end{matrix}\right)\left(\begin{matrix}26\\53\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{17}\times 26-\frac{1}{17}\times 53\\-\frac{3}{34}\times 26+\frac{5}{34}\times 53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\\frac{11}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=\frac{11}{2}$$

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.

$$5x+2y=26,3x+8y=53$$

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3\times 2y=3\times 26,5\times 3x+5\times 8y=5\times 53$$

Whakarūnātia.

$$15x+6y=78,15x+40y=265$$

Me tango 15x+40y=265 mai i 15x+6y=78 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+6y-40y=78-265$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-40y=78-265$$

Tāpiri 6y ki te -40y.

$$-34y=78-265$$

Tāpiri 78 ki te -265.

$$-34y=-187$$

Whakawehea ngā taha e rua ki te -34.

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

Whakaurua te \frac{11}{2} mō y ki 3x+8y=53. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+8\times \frac{11}{2}=53$$

Whakareatia 8 ki te \frac{11}{2}.

$$3x+44=53$$

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

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{11}{2}$$