Whakaoti i te {l}{-5x+13y=-7}{5x+4y=24}

$$\left. \begin{array} { l } { - 5 x + 13 y = - 7 } \\ { 5 x + 4 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=1$$

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.

$$-5x+13y=-7,5x+4y=24$$

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

$$-5x+13y=-7$$

Me tango 13y mai i ngā taha e rua o te whārite.

$$-5x=-13y-7$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-13y-7\right)$$

Whakareatia -\frac{1}{5} ki te -13y-7.

$$x=\frac{13}{5}y+\frac{7}{5}$$

Whakakapia te \frac{13y+7}{5} mō te x ki tērā atu whārite, 5x+4y=24.

$$5\left(\frac{13}{5}y+\frac{7}{5}\right)+4y=24$$

Whakareatia 5 ki te \frac{13y+7}{5}.

$$13y+7+4y=24$$

Tāpiri 13y ki te 4y.

$$17y+7=24$$

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

$$17y=17$$

Whakawehea ngā taha e rua ki te 17.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{13}{5}y+\frac{7}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{13+7}{5}$$

Tāpiri \frac{7}{5} ki te \frac{13}{5} 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.

$$x=4,y=1$$

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.

$$-5x+13y=-7,5x+4y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&13\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&13\\5&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&13\\5&4\end{matrix}\right))\left(\begin{matrix}-5&13\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&13\\5&4\end{matrix}\right))\left(\begin{matrix}-7\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&13\\5&4\end{matrix}\right))\left(\begin{matrix}-7\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&13\\5&4\end{matrix}\right))\left(\begin{matrix}-7\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{-5\times 4-13\times 5}&-\frac{13}{-5\times 4-13\times 5}\\-\frac{5}{-5\times 4-13\times 5}&-\frac{5}{-5\times 4-13\times 5}\end{matrix}\right)\left(\begin{matrix}-7\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{85}&\frac{13}{85}\\\frac{1}{17}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}-7\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{85}\left(-7\right)+\frac{13}{85}\times 24\\\frac{1}{17}\left(-7\right)+\frac{1}{17}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=1$$

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.

$$-5x+13y=-7,5x+4y=24$$

Kia ōrite ai a -5x 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 -5.

$$5\left(-5\right)x+5\times 13y=5\left(-7\right),-5\times 5x-5\times 4y=-5\times 24$$

Whakarūnātia.

$$-25x+65y=-35,-25x-20y=-120$$

Me tango -25x-20y=-120 mai i -25x+65y=-35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-25x+25x+65y+20y=-35+120$$

Tāpiri -25x ki te 25x. Ka whakakore atu ngā kupu -25x me 25x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$65y+20y=-35+120$$

Tāpiri 65y ki te 20y.

$$85y=-35+120$$

Tāpiri -35 ki te 120.

$$85y=85$$

Whakawehea ngā taha e rua ki te 85.

$$y=1$$

Whakaurua te 1 mō y ki 5x+4y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+4=24$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=1$$