Whakaoti i te {l}{-10x-7y=-5}{7x+5y=4}

$$\left. \begin{array} { l } { - 10 x - 7 y = - 5 } \\ { 7 x + 5 y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=5$$

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.

$$-10x-7y=-5,7x+5y=4$$

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

$$-10x-7y=-5$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$-10x=7y-5$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(7y-5\right)$$

Whakareatia -\frac{1}{10} ki te 7y-5.

$$x=-\frac{7}{10}y+\frac{1}{2}$$

Whakakapia te -\frac{7y}{10}+\frac{1}{2} mō te x ki tērā atu whārite, 7x+5y=4.

$$7\left(-\frac{7}{10}y+\frac{1}{2}\right)+5y=4$$

Whakareatia 7 ki te -\frac{7y}{10}+\frac{1}{2}.

$$-\frac{49}{10}y+\frac{7}{2}+5y=4$$

Tāpiri -\frac{49y}{10} ki te 5y.

$$\frac{1}{10}y+\frac{7}{2}=4$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{10}y=\frac{1}{2}$$

Me whakarea ngā taha e rua ki te 10.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{7}{10}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{10}\times 5+\frac{1}{2}$$

Whakareatia -\frac{7}{10} ki te 5.

$$x=\frac{-7+1}{2}$$

Tāpiri \frac{1}{2} ki te -\frac{7}{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.

$$x=-3,y=5$$

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.

$$-10x-7y=-5,7x+5y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&-7\\7&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right))\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right))\left(\begin{matrix}-5\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right))\left(\begin{matrix}-5\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&-7\\7&5\end{matrix}\right))\left(\begin{matrix}-5\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-10\times 5-\left(-7\times 7\right)}&-\frac{-7}{-10\times 5-\left(-7\times 7\right)}\\-\frac{7}{-10\times 5-\left(-7\times 7\right)}&-\frac{10}{-10\times 5-\left(-7\times 7\right)}\end{matrix}\right)\left(\begin{matrix}-5\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5&-7\\7&10\end{matrix}\right)\left(\begin{matrix}-5\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\left(-5\right)-7\times 4\\7\left(-5\right)+10\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=5$$

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.

$$-10x-7y=-5,7x+5y=4$$

Kia ōrite ai a -10x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -10.

$$7\left(-10\right)x+7\left(-7\right)y=7\left(-5\right),-10\times 7x-10\times 5y=-10\times 4$$

Whakarūnātia.

$$-70x-49y=-35,-70x-50y=-40$$

Me tango -70x-50y=-40 mai i -70x-49y=-35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-70x+70x-49y+50y=-35+40$$

Tāpiri -70x ki te 70x. Ka whakakore atu ngā kupu -70x me 70x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-49y+50y=-35+40$$

Tāpiri -49y ki te 50y.

$$y=-35+40$$

Tāpiri -35 ki te 40.

$$y=5$$

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

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

Whakareatia 5 ki te 5.

$$7x+25=4$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$7x=-21$$

Whakawehea ngā taha e rua ki te 7.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=5$$