Whakaoti i te {l}{5x-3y=28}{12x+4y=0}

$$\left\{ \begin{array} { l } { 5 x - 3 y = 28 } \\ { 12 x + 4 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-6$$

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.

$$5x-3y=28,12x+4y=0$$

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

$$5x-3y=28$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+28$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+28\right)$$

Whakareatia \frac{1}{5} ki te 3y+28.

$$x=\frac{3}{5}y+\frac{28}{5}$$

Whakakapia te \frac{3y+28}{5} mō te x ki tērā atu whārite, 12x+4y=0.

$$12\left(\frac{3}{5}y+\frac{28}{5}\right)+4y=0$$

Whakareatia 12 ki te \frac{3y+28}{5}.

$$\frac{36}{5}y+\frac{336}{5}+4y=0$$

Tāpiri \frac{36y}{5} ki te 4y.

$$\frac{56}{5}y+\frac{336}{5}=0$$

Me tango \frac{336}{5} mai i ngā taha e rua o te whārite.

$$\frac{56}{5}y=-\frac{336}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{56}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

Whakaurua te -6 mō y ki x=\frac{3}{5}y+\frac{28}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-6\right)+\frac{28}{5}$$

Whakareatia \frac{3}{5} ki te -6.

$$x=\frac{-18+28}{5}$$

Tāpiri \frac{28}{5} ki te -\frac{18}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-6$$

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.

$$5x-3y=28,12x+4y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\12&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\12&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\12&4\end{matrix}\right))\left(\begin{matrix}5&-3\\12&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\12&4\end{matrix}\right))\left(\begin{matrix}28\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\12&4\end{matrix}\right))\left(\begin{matrix}28\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\12&4\end{matrix}\right))\left(\begin{matrix}28\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-\left(-3\times 12\right)}&-\frac{-3}{5\times 4-\left(-3\times 12\right)}\\-\frac{12}{5\times 4-\left(-3\times 12\right)}&\frac{5}{5\times 4-\left(-3\times 12\right)}\end{matrix}\right)\left(\begin{matrix}28\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{3}{56}\\-\frac{3}{14}&\frac{5}{56}\end{matrix}\right)\left(\begin{matrix}28\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\times 28\\-\frac{3}{14}\times 28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-6$$

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.

$$5x-3y=28,12x+4y=0$$

Kia ōrite ai a 5x me 12x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$12\times 5x+12\left(-3\right)y=12\times 28,5\times 12x+5\times 4y=0$$

Whakarūnātia.

$$60x-36y=336,60x+20y=0$$

Me tango 60x+20y=0 mai i 60x-36y=336 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$60x-60x-36y-20y=336$$

Tāpiri 60x ki te -60x. Ka whakakore atu ngā kupu 60x me -60x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-36y-20y=336$$

Tāpiri -36y ki te -20y.

$$-56y=336$$

Whakawehea ngā taha e rua ki te -56.

$$y=-6$$

Whakaurua te -6 mō y ki 12x+4y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$12x+4\left(-6\right)=0$$

Whakareatia 4 ki te -6.

$$12x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$12x=24$$

Whakawehea ngā taha e rua ki te 12.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-6$$