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

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

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.

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

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+y=15.

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

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

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

Tāpiri \frac{15y}{2} ki te y.

$$\frac{17}{2}y-\frac{21}{2}=15$$

Me tāpiri \frac{21}{2} ki ngā taha e rua o te whārite.

$$\frac{17}{2}y=\frac{51}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{2}, 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{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.

$$x=-\frac{5}{2}\times 3+\frac{7}{2}$$

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

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

Tāpiri \frac{7}{2} ki te -\frac{15}{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.

$$x=-4$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\-3&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}&-\frac{5}{17}\\\frac{3}{17}&\frac{2}{17}\end{matrix}\right)\left(\begin{matrix}7\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}\times 7-\frac{5}{17}\times 15\\\frac{3}{17}\times 7+\frac{2}{17}\times 15\end{matrix}\right)$$

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.

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

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.

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

Whakarūnātia.

$$-6x-15y=-21,-6x+2y=30$$

Me tango -6x+2y=30 mai i -6x-15y=-21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x-15y-2y=-21-30$$

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.

$$-15y-2y=-21-30$$

Tāpiri -15y ki te -2y.

$$-17y=-21-30$$

Tāpiri -21 ki te -30.

$$-17y=-51$$

Whakawehea ngā taha e rua ki te -17.

$$y=3$$

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

$$-3x+3=15$$

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

$$-3x=12$$

Whakawehea ngā taha e rua ki te -3.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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