Whakaoti i te {l}{2x+7y=15}{3x-5y=23}

$$\left\{ \begin{array} { l } { 2 x + 7 y = 15 } \\ { 3 x - 5 y = 23 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{236}{31} = 7\frac{19}{31} \approx 7.612903226$$

$$y=-\frac{1}{31}\approx -0.032258065$$

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.

$$2x+7y=15,3x-5y=23$$

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

$$2x+7y=15$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$2x=-7y+15$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-7y+15\right)$$

Whakareatia \frac{1}{2} ki te -7y+15.

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

Whakakapia te \frac{-7y+15}{2} mō te x ki tērā atu whārite, 3x-5y=23.

$$3\left(-\frac{7}{2}y+\frac{15}{2}\right)-5y=23$$

Whakareatia 3 ki te \frac{-7y+15}{2}.

$$-\frac{21}{2}y+\frac{45}{2}-5y=23$$

Tāpiri -\frac{21y}{2} ki te -5y.

$$-\frac{31}{2}y+\frac{45}{2}=23$$

Me tango \frac{45}{2} mai i ngā taha e rua o te whārite.

$$-\frac{31}{2}y=\frac{1}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{31}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{1}{31}$$

Whakaurua te -\frac{1}{31} mō y ki x=-\frac{7}{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.

$$x=-\frac{7}{2}\left(-\frac{1}{31}\right)+\frac{15}{2}$$

Whakareatia -\frac{7}{2} ki te -\frac{1}{31} 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.

$$x=\frac{7}{62}+\frac{15}{2}$$

Tāpiri \frac{15}{2} ki te \frac{7}{62} 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.

$$x=\frac{236}{31}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{236}{31},y=-\frac{1}{31}$$

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.

$$2x+7y=15,3x-5y=23$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&7\\3&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\23\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&7\\3&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\3&-5\end{matrix}\right))\left(\begin{matrix}2&7\\3&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&-5\end{matrix}\right))\left(\begin{matrix}15\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&-5\end{matrix}\right))\left(\begin{matrix}15\\23\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\3&-5\end{matrix}\right))\left(\begin{matrix}15\\23\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2\left(-5\right)-7\times 3}&-\frac{7}{2\left(-5\right)-7\times 3}\\-\frac{3}{2\left(-5\right)-7\times 3}&\frac{2}{2\left(-5\right)-7\times 3}\end{matrix}\right)\left(\begin{matrix}15\\23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}&\frac{7}{31}\\\frac{3}{31}&-\frac{2}{31}\end{matrix}\right)\left(\begin{matrix}15\\23\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}\times 15+\frac{7}{31}\times 23\\\frac{3}{31}\times 15-\frac{2}{31}\times 23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{236}{31}\\-\frac{1}{31}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{236}{31},y=-\frac{1}{31}$$

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.

$$2x+7y=15,3x-5y=23$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\times 7y=3\times 15,2\times 3x+2\left(-5\right)y=2\times 23$$

Whakarūnātia.

$$6x+21y=45,6x-10y=46$$

Me tango 6x-10y=46 mai i 6x+21y=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+21y+10y=45-46$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y+10y=45-46$$

Tāpiri 21y ki te 10y.

$$31y=45-46$$

Tāpiri 45 ki te -46.

$$31y=-1$$

Whakawehea ngā taha e rua ki te 31.

$$y=-\frac{1}{31}$$

Whakaurua te -\frac{1}{31} mō y ki 3x-5y=23. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-5\left(-\frac{1}{31}\right)=23$$

Whakareatia -5 ki te -\frac{1}{31}.

$$3x+\frac{5}{31}=23$$

Me tango \frac{5}{31} mai i ngā taha e rua o te whārite.

$$3x=\frac{708}{31}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{236}{31}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{236}{31},y=-\frac{1}{31}$$