Whakaoti i te {l}{3x+5y/6=-5}{2(x+7)+3y=-5}

$$\left\{ \begin{array} { l } { \frac { 3 x + 5 y } { 6 } = - 5 } \\ { 2 ( x + 7 ) + 3 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 6.

$$3x+5y=-5\times 6$$

Whakareatia te -5 ki te 6, ka -30.

$$3x+5y=-30$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+7.

$$2x+14+3y=-5$$

Tangohia te 14 mai i ngā taha e rua.

$$2x+3y=-5-14$$

Tangohia te 14 i te -5, ka -19.

$$2x+3y=-19$$

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.

$$3x+5y=-30,2x+3y=-19$$

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

$$3x+5y=-30$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y-30$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y-30\right)$$

Whakareatia \frac{1}{3} ki te -5y-30.

$$x=-\frac{5}{3}y-10$$

Whakakapia te -\frac{5y}{3}-10 mō te x ki tērā atu whārite, 2x+3y=-19.

$$2\left(-\frac{5}{3}y-10\right)+3y=-19$$

Whakareatia 2 ki te -\frac{5y}{3}-10.

$$-\frac{10}{3}y-20+3y=-19$$

Tāpiri -\frac{10y}{3} ki te 3y.

$$-\frac{1}{3}y-20=-19$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$-\frac{1}{3}y=1$$

Me whakarea ngā taha e rua ki te -3.

$$y=-3$$

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

$$x=-\frac{5}{3}\left(-3\right)-10$$

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

$$x=5-10$$

Tāpiri -10 ki te 5.

$$x=-5$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 6.

$$3x+5y=-5\times 6$$

Whakareatia te -5 ki te 6, ka -30.

$$3x+5y=-30$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+7.

$$2x+14+3y=-5$$

Tangohia te 14 mai i ngā taha e rua.

$$2x+3y=-5-14$$

Tangohia te 14 i te -5, ka -19.

$$2x+3y=-19$$

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

$$3x+5y=-30,2x+3y=-19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}3&5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\2&3\end{matrix}\right))\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-5\times 2}&-\frac{5}{3\times 3-5\times 2}\\-\frac{2}{3\times 3-5\times 2}&\frac{3}{3\times 3-5\times 2}\end{matrix}\right)\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&5\\2&-3\end{matrix}\right)\left(\begin{matrix}-30\\-19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\left(-30\right)+5\left(-19\right)\\2\left(-30\right)-3\left(-19\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 6.

$$3x+5y=-5\times 6$$

Whakareatia te -5 ki te 6, ka -30.

$$3x+5y=-30$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+7.

$$2x+14+3y=-5$$

Tangohia te 14 mai i ngā taha e rua.

$$2x+3y=-5-14$$

Tangohia te 14 i te -5, ka -19.

$$2x+3y=-19$$

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.

$$3x+5y=-30,2x+3y=-19$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\times 5y=2\left(-30\right),3\times 2x+3\times 3y=3\left(-19\right)$$

Whakarūnātia.

$$6x+10y=-60,6x+9y=-57$$

Me tango 6x+9y=-57 mai i 6x+10y=-60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+10y-9y=-60+57$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-9y=-60+57$$

Tāpiri 10y ki te -9y.

$$y=-60+57$$

Tāpiri -60 ki te 57.

$$y=-3$$

Whakaurua te -3 mō y ki 2x+3y=-19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-3\right)=-19$$

Whakareatia 3 ki te -3.

$$2x-9=-19$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$2x=-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

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