Whakaoti i te {l}{2x+3y=15}{5x+4y=13}

$$\left. \begin{array} { l } { 2 x + 3 y = 15 } \\ { 5 x + 4 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=7$$

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.

$$2x+3y=15,5x+4y=13$$

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

$$2x+3y=15$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+15$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+15\right)$$

Whakareatia \frac{1}{2} ki te -3y+15.

$$x=-\frac{3}{2}y+\frac{15}{2}$$

Whakakapia te \frac{-3y+15}{2} mō te x ki tērā atu whārite, 5x+4y=13.

$$5\left(-\frac{3}{2}y+\frac{15}{2}\right)+4y=13$$

Whakareatia 5 ki te \frac{-3y+15}{2}.

$$-\frac{15}{2}y+\frac{75}{2}+4y=13$$

Tāpiri -\frac{15y}{2} ki te 4y.

$$-\frac{7}{2}y+\frac{75}{2}=13$$

Me tango \frac{75}{2} mai i ngā taha e rua o te whārite.

$$-\frac{7}{2}y=-\frac{49}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{3}{2}y+\frac{15}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 7+\frac{15}{2}$$

Whakareatia -\frac{3}{2} ki te 7.

$$x=\frac{-21+15}{2}$$

Tāpiri \frac{15}{2} ki te -\frac{21}{2} 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.

$$x=-3,y=7$$

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.

$$2x+3y=15,5x+4y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\13\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\5&4\end{matrix}\right))\left(\begin{matrix}2&3\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&4\end{matrix}\right))\left(\begin{matrix}15\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&4\end{matrix}\right))\left(\begin{matrix}15\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\5&4\end{matrix}\right))\left(\begin{matrix}15\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{2\times 4-3\times 5}&-\frac{3}{2\times 4-3\times 5}\\-\frac{5}{2\times 4-3\times 5}&\frac{2}{2\times 4-3\times 5}\end{matrix}\right)\left(\begin{matrix}15\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}&\frac{3}{7}\\\frac{5}{7}&-\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}15\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}\times 15+\frac{3}{7}\times 13\\\frac{5}{7}\times 15-\frac{2}{7}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=7$$

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.

$$2x+3y=15,5x+4y=13$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\times 3y=5\times 15,2\times 5x+2\times 4y=2\times 13$$

Whakarūnātia.

$$10x+15y=75,10x+8y=26$$

Me tango 10x+8y=26 mai i 10x+15y=75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+15y-8y=75-26$$

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.

$$15y-8y=75-26$$

Tāpiri 15y ki te -8y.

$$7y=75-26$$

Tāpiri 75 ki te -26.

$$7y=49$$

Whakawehea ngā taha e rua ki te 7.

$$y=7$$

Whakaurua te 7 mō y ki 5x+4y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+4\times 7=13$$

Whakareatia 4 ki te 7.

$$5x+28=13$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$5x=-15$$

Whakawehea ngā taha e rua ki te 5.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=7$$