Whakaoti i te {l}{3y-6=x}{x-9=2y}

$$\left. \begin{array} { l } { 3 y - 6 = x } \\ { x - 9 = 2 y } \end{array} \right.$$

Whakaoti mō y, x

$$x=39$$

$$y=15$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$3y-6-x=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-x=6$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$x-9-2y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-2y=9$$

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.

$$3y-x=6,-2y+x=9$$

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

$$3y-x=6$$

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

$$3y=x+6$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(x+6\right)$$

Whakareatia \frac{1}{3} ki te x+6.

$$y=\frac{1}{3}x+2$$

Whakakapia te \frac{x}{3}+2 mō te y ki tērā atu whārite, -2y+x=9.

$$-2\left(\frac{1}{3}x+2\right)+x=9$$

Whakareatia -2 ki te \frac{x}{3}+2.

$$-\frac{2}{3}x-4+x=9$$

Tāpiri -\frac{2x}{3} ki te x.

$$\frac{1}{3}x-4=9$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$\frac{1}{3}x=13$$

Me whakarea ngā taha e rua ki te 3.

$$x=39$$

Whakaurua te 39 mō x ki y=\frac{1}{3}x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{3}\times 39+2$$

Whakareatia \frac{1}{3} ki te 39.

$$y=13+2$$

Tāpiri 2 ki te 13.

$$y=15$$

Kua oti te pūnaha te whakatau.

$$y=15,x=39$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3y-6-x=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-x=6$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$x-9-2y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-2y=9$$

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

$$3y-x=6,-2y+x=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\-2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right))\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-\left(-2\right)\right)}&-\frac{-1}{3-\left(-\left(-2\right)\right)}\\-\frac{-2}{3-\left(-\left(-2\right)\right)}&\frac{3}{3-\left(-\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6+9\\2\times 6+3\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}15\\39\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=15,x=39$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$3y-6-x=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-x=6$$

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$x-9-2y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-2y=9$$

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.

$$3y-x=6,-2y+x=9$$

Kia ōrite ai a 3y me -2y, 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.

$$-2\times 3y-2\left(-1\right)x=-2\times 6,3\left(-2\right)y+3x=3\times 9$$

Whakarūnātia.

$$-6y+2x=-12,-6y+3x=27$$

Me tango -6y+3x=27 mai i -6y+2x=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6y+6y+2x-3x=-12-27$$

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

$$2x-3x=-12-27$$

Tāpiri 2x ki te -3x.

$$-x=-12-27$$

Tāpiri -12 ki te -27.

$$-x=-39$$

Whakawehea ngā taha e rua ki te -1.

$$x=39$$

Whakaurua te 39 mō x ki -2y+x=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2y+39=9$$

Me tango 39 mai i ngā taha e rua o te whārite.

$$-2y=-30$$

Whakawehea ngā taha e rua ki te -2.

$$y=15$$

Kua oti te pūnaha te whakatau.

$$y=15,x=39$$