Whakaoti i te {r}{7x+3y=-15}{12y-5x=39}

$$\left\{ \begin{array} { r } { 7 x + 3 y = - 15 } \\ { 12 y - 5 x = 39 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=2$$

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.

$$7x+3y=-15,-5x+12y=39$$

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

$$7x+3y=-15$$

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

$$7x=-3y-15$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-3y-15\right)$$

Whakareatia \frac{1}{7} ki te -3y-15.

$$x=-\frac{3}{7}y-\frac{15}{7}$$

Whakakapia te \frac{-3y-15}{7} mō te x ki tērā atu whārite, -5x+12y=39.

$$-5\left(-\frac{3}{7}y-\frac{15}{7}\right)+12y=39$$

Whakareatia -5 ki te \frac{-3y-15}{7}.

$$\frac{15}{7}y+\frac{75}{7}+12y=39$$

Tāpiri \frac{15y}{7} ki te 12y.

$$\frac{99}{7}y+\frac{75}{7}=39$$

Me tango \frac{75}{7} mai i ngā taha e rua o te whārite.

$$\frac{99}{7}y=\frac{198}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{99}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

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

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

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

$$x=\frac{-6-15}{7}$$

Tāpiri -\frac{15}{7} ki te -\frac{6}{7} 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.

$$x=-3,y=2$$

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.

$$7x+3y=-15,-5x+12y=39$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&3\\-5&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-15\\39\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&3\\-5&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&3\\-5&12\end{matrix}\right))\left(\begin{matrix}7&3\\-5&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\-5&12\end{matrix}\right))\left(\begin{matrix}-15\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\-5&12\end{matrix}\right))\left(\begin{matrix}-15\\39\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\-5&12\end{matrix}\right))\left(\begin{matrix}-15\\39\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{7\times 12-3\left(-5\right)}&-\frac{3}{7\times 12-3\left(-5\right)}\\-\frac{-5}{7\times 12-3\left(-5\right)}&\frac{7}{7\times 12-3\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}-15\\39\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{33}&-\frac{1}{33}\\\frac{5}{99}&\frac{7}{99}\end{matrix}\right)\left(\begin{matrix}-15\\39\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{33}\left(-15\right)-\frac{1}{33}\times 39\\\frac{5}{99}\left(-15\right)+\frac{7}{99}\times 39\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=2$$

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.

$$7x+3y=-15,-5x+12y=39$$

Kia ōrite ai a 7x 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 7.

$$-5\times 7x-5\times 3y=-5\left(-15\right),7\left(-5\right)x+7\times 12y=7\times 39$$

Whakarūnātia.

$$-35x-15y=75,-35x+84y=273$$

Me tango -35x+84y=273 mai i -35x-15y=75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-35x+35x-15y-84y=75-273$$

Tāpiri -35x ki te 35x. Ka whakakore atu ngā kupu -35x me 35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-84y=75-273$$

Tāpiri -15y ki te -84y.

$$-99y=75-273$$

Tāpiri 75 ki te -273.

$$-99y=-198$$

Whakawehea ngā taha e rua ki te -99.

$$y=2$$

Whakaurua te 2 mō y ki -5x+12y=39. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+12\times 2=39$$

Whakareatia 12 ki te 2.

$$-5x+24=39$$

Me tango 24 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.

$$x=-3,y=2$$