Whakaoti i te {l}{x/2+y/6=11/2}{2x/5-frac{y}{3}=-frac{1}{5}}

$$\left\{ \begin{array} { l } { \frac { x } { 2 } + \frac { y } { 6 } = 1 \frac { 1 } { 2 } } \\ { \frac { 2 x } { 5 } - \frac { y } { 3 } = - \frac { 1 } { 5 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,6.

$$3x+y=3\left(1\times 2+1\right)$$

Whakareatia te 1 ki te 2, ka 2.

$$3x+y=3\left(2+1\right)$$

Tāpirihia te 2 ki te 1, ka 3.

$$3x+y=3\times 3$$

Whakareatia te 3 ki te 3, ka 9.

$$3x+y=9$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3\times 2x-5y=-3$$

Whakareatia te 3 ki te 2, ka 6.

$$6x-5y=-3$$

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.

$$3x+y=9,6x-5y=-3$$

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

$$3x+y=9$$

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

$$3x=-y+9$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te -y+9.

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

Whakakapia te -\frac{y}{3}+3 mō te x ki tērā atu whārite, 6x-5y=-3.

$$6\left(-\frac{1}{3}y+3\right)-5y=-3$$

Whakareatia 6 ki te -\frac{y}{3}+3.

$$-2y+18-5y=-3$$

Tāpiri -2y ki te -5y.

$$-7y+18=-3$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-7y=-21$$

Whakawehea ngā taha e rua ki te -7.

$$y=3$$

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

$$x=-\frac{1}{3}\times 3+3$$

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

$$x=-1+3$$

Tāpiri 3 ki te -1.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,6.

$$3x+y=3\left(1\times 2+1\right)$$

Whakareatia te 1 ki te 2, ka 2.

$$3x+y=3\left(2+1\right)$$

Tāpirihia te 2 ki te 1, ka 3.

$$3x+y=3\times 3$$

Whakareatia te 3 ki te 3, ka 9.

$$3x+y=9$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3\times 2x-5y=-3$$

Whakareatia te 3 ki te 2, ka 6.

$$6x-5y=-3$$

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

$$3x+y=9,6x-5y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\6&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\6&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\6&-5\end{matrix}\right))\left(\begin{matrix}3&1\\6&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&-5\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&-5\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&-5\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3\left(-5\right)-6}&-\frac{1}{3\left(-5\right)-6}\\-\frac{6}{3\left(-5\right)-6}&\frac{3}{3\left(-5\right)-6}\end{matrix}\right)\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}&\frac{1}{21}\\\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\times 9+\frac{1}{21}\left(-3\right)\\\frac{2}{7}\times 9-\frac{1}{7}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,6.

$$3x+y=3\left(1\times 2+1\right)$$

Whakareatia te 1 ki te 2, ka 2.

$$3x+y=3\left(2+1\right)$$

Tāpirihia te 2 ki te 1, ka 3.

$$3x+y=3\times 3$$

Whakareatia te 3 ki te 3, ka 9.

$$3x+y=9$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3\times 2x-5y=-3$$

Whakareatia te 3 ki te 2, ka 6.

$$6x-5y=-3$$

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.

$$3x+y=9,6x-5y=-3$$

Kia ōrite ai a 3x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$6\times 3x+6y=6\times 9,3\times 6x+3\left(-5\right)y=3\left(-3\right)$$

Whakarūnātia.

$$18x+6y=54,18x-15y=-9$$

Me tango 18x-15y=-9 mai i 18x+6y=54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+6y+15y=54+9$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+15y=54+9$$

Tāpiri 6y ki te 15y.

$$21y=54+9$$

Tāpiri 54 ki te 9.

$$21y=63$$

Whakawehea ngā taha e rua ki te 21.

$$y=3$$

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

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

Whakareatia -5 ki te 3.

$$6x-15=-3$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$6x=12$$

Whakawehea ngā taha e rua ki te 6.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$