Whakaoti i te {l}{2x+5y=7}{3x+7y=10}

$$\left. \begin{array} { l } { 2 x + 5 y = 7 } \\ { 3 x + 7 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

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.

$$2x+5y=7,3x+7y=10$$

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

$$2x+5y=7$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y+7$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+7\right)$$

Whakareatia \frac{1}{2} ki te -5y+7.

$$x=-\frac{5}{2}y+\frac{7}{2}$$

Whakakapia te \frac{-5y+7}{2} mō te x ki tērā atu whārite, 3x+7y=10.

$$3\left(-\frac{5}{2}y+\frac{7}{2}\right)+7y=10$$

Whakareatia 3 ki te \frac{-5y+7}{2}.

$$-\frac{15}{2}y+\frac{21}{2}+7y=10$$

Tāpiri -\frac{15y}{2} ki te 7y.

$$-\frac{1}{2}y+\frac{21}{2}=10$$

Me tango \frac{21}{2} mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te -2.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{5}{2}y+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-5+7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{5}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$

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.

$$2x+5y=7,3x+7y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&5\\3&7\end{matrix}\right))\left(\begin{matrix}2&5\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&7\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&7\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&7\end{matrix}\right))\left(\begin{matrix}7\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-5\times 3}&-\frac{5}{2\times 7-5\times 3}\\-\frac{3}{2\times 7-5\times 3}&\frac{2}{2\times 7-5\times 3}\end{matrix}\right)\left(\begin{matrix}7\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7&5\\3&-2\end{matrix}\right)\left(\begin{matrix}7\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\times 7+5\times 10\\3\times 7-2\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

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.

$$2x+5y=7,3x+7y=10$$

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.

$$3\times 2x+3\times 5y=3\times 7,2\times 3x+2\times 7y=2\times 10$$

Whakarūnātia.

$$6x+15y=21,6x+14y=20$$

Me tango 6x+14y=20 mai i 6x+15y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+15y-14y=21-20$$

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.

$$15y-14y=21-20$$

Tāpiri 15y ki te -14y.

$$y=21-20$$

Tāpiri 21 ki te -20.

$$y=1$$

Whakaurua te 1 mō y ki 3x+7y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+7=10$$

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

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$