Whakaoti i te {l}{-5x+y=-12}{5y=10x-15}

$$\left\{ \begin{array} { l } { - 5 x + y = - 12 } \\ { 5 y = 10 x - 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$5y-10x=-15$$

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.

$$-5x+y=-12,-10x+5y=-15$$

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

$$-5x+y=-12$$

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

$$-5x=-y-12$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-y-12\right)$$

Whakareatia -\frac{1}{5} ki te -y-12.

$$x=\frac{1}{5}y+\frac{12}{5}$$

Whakakapia te \frac{12+y}{5} mō te x ki tērā atu whārite, -10x+5y=-15.

$$-10\left(\frac{1}{5}y+\frac{12}{5}\right)+5y=-15$$

Whakareatia -10 ki te \frac{12+y}{5}.

$$-2y-24+5y=-15$$

Tāpiri -2y ki te 5y.

$$3y-24=-15$$

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

$$3y=9$$

Whakawehea ngā taha e rua ki te 3.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{1}{5}y+\frac{12}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times 3+\frac{12}{5}$$

Whakareatia \frac{1}{5} ki te 3.

$$x=\frac{3+12}{5}$$

Tāpiri \frac{12}{5} ki te \frac{3}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$5y-10x=-15$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-5x+y=-12,-10x+5y=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&1\\-10&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right))\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&1\\-10&5\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-5\times 5-\left(-10\right)}&-\frac{1}{-5\times 5-\left(-10\right)}\\-\frac{-10}{-5\times 5-\left(-10\right)}&-\frac{5}{-5\times 5-\left(-10\right)}\end{matrix}\right)\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{15}\\-\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-12\right)+\frac{1}{15}\left(-15\right)\\-\frac{2}{3}\left(-12\right)+\frac{1}{3}\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$5y-10x=-15$$

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.

$$-5x+y=-12,-10x+5y=-15$$

Kia ōrite ai a -5x me -10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$-10\left(-5\right)x-10y=-10\left(-12\right),-5\left(-10\right)x-5\times 5y=-5\left(-15\right)$$

Whakarūnātia.

$$50x-10y=120,50x-25y=75$$

Me tango 50x-25y=75 mai i 50x-10y=120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x-10y+25y=120-75$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y+25y=120-75$$

Tāpiri -10y ki te 25y.

$$15y=120-75$$

Tāpiri 120 ki te -75.

$$15y=45$$

Whakawehea ngā taha e rua ki te 15.

$$y=3$$

Whakaurua te 3 mō y ki -10x+5y=-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-10x+5\times 3=-15$$

Whakareatia 5 ki te 3.

$$-10x+15=-15$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-10x=-30$$

Whakawehea ngā taha e rua ki te -10.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=3$$