Whakaoti i te {l}{5x+4y=8}{2x-3y=17}

$$\left. \begin{array} { l } { 5 x + 4 y = 8 } \\ { 2 x - 3 y = 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-3$$

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.

$$5x+4y=8,2x-3y=17$$

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

$$5x+4y=8$$

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

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

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-4y+8\right)$$

Whakareatia \frac{1}{5} ki te -4y+8.

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

Whakakapia te \frac{-4y+8}{5} mō te x ki tērā atu whārite, 2x-3y=17.

$$2\left(-\frac{4}{5}y+\frac{8}{5}\right)-3y=17$$

Whakareatia 2 ki te \frac{-4y+8}{5}.

$$-\frac{8}{5}y+\frac{16}{5}-3y=17$$

Tāpiri -\frac{8y}{5} ki te -3y.

$$-\frac{23}{5}y+\frac{16}{5}=17$$

Me tango \frac{16}{5} mai i ngā taha e rua o te whārite.

$$-\frac{23}{5}y=\frac{69}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{23}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 mō y ki x=-\frac{4}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia -\frac{4}{5} ki te -3.

$$x=\frac{12+8}{5}$$

Tāpiri \frac{8}{5} ki te \frac{12}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-3$$

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.

$$5x+4y=8,2x-3y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&4\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&4\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&4\\2&-3\end{matrix}\right))\left(\begin{matrix}5&4\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&-3\end{matrix}\right))\left(\begin{matrix}8\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&-3\end{matrix}\right))\left(\begin{matrix}8\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&-3\end{matrix}\right))\left(\begin{matrix}8\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-4\times 2}&-\frac{4}{5\left(-3\right)-4\times 2}\\-\frac{2}{5\left(-3\right)-4\times 2}&\frac{5}{5\left(-3\right)-4\times 2}\end{matrix}\right)\left(\begin{matrix}8\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{23}&\frac{4}{23}\\\frac{2}{23}&-\frac{5}{23}\end{matrix}\right)\left(\begin{matrix}8\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{23}\times 8+\frac{4}{23}\times 17\\\frac{2}{23}\times 8-\frac{5}{23}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-3$$

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.

$$5x+4y=8,2x-3y=17$$

Kia ōrite ai a 5x 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 5.

$$2\times 5x+2\times 4y=2\times 8,5\times 2x+5\left(-3\right)y=5\times 17$$

Whakarūnātia.

$$10x+8y=16,10x-15y=85$$

Me tango 10x-15y=85 mai i 10x+8y=16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+8y+15y=16-85$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y+15y=16-85$$

Tāpiri 8y ki te 15y.

$$23y=16-85$$

Tāpiri 16 ki te -85.

$$23y=-69$$

Whakawehea ngā taha e rua ki te 23.

$$y=-3$$

Whakaurua te -3 mō y ki 2x-3y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-3\right)=17$$

Whakareatia -3 ki te -3.

$$2x+9=17$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-3$$