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

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

Whakaoti mō x, y

$$x=2$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x+1\right)+6=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2+6=3\left(5-y\right)$$

Tāpirihia te 2 ki te 6, ka 8.

$$2x+8=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5-y.

$$2x+8=15-3y$$

Me tāpiri te 3y ki ngā taha e rua.

$$2x+8+3y=15$$

Tangohia te 8 mai i ngā taha e rua.

$$2x+3y=15-8$$

Tangohia te 8 i te 15, ka 7.

$$2x+3y=7$$

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.

$$2x-3y=1,2x+3y=7$$

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

$$2x-3y=1$$

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

$$2x=3y+1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 3y+1.

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

Whakakapia te \frac{3y+1}{2} mō te x ki tērā atu whārite, 2x+3y=7.

$$2\left(\frac{3}{2}y+\frac{1}{2}\right)+3y=7$$

Whakareatia 2 ki te \frac{3y+1}{2}.

$$3y+1+3y=7$$

Tāpiri 3y ki te 3y.

$$6y+1=7$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$6y=6$$

Whakawehea ngā taha e rua ki te 6.

$$y=1$$

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

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

Tāpiri \frac{1}{2} ki te \frac{3}{2} 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.

$$x=2,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x+1\right)+6=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2+6=3\left(5-y\right)$$

Tāpirihia te 2 ki te 6, ka 8.

$$2x+8=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5-y.

$$2x+8=15-3y$$

Me tāpiri te 3y ki ngā taha e rua.

$$2x+8+3y=15$$

Tangohia te 8 mai i ngā taha e rua.

$$2x+3y=15-8$$

Tangohia te 8 i te 15, ka 7.

$$2x+3y=7$$

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

$$2x-3y=1,2x+3y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&3\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}+\frac{1}{4}\times 7\\-\frac{1}{6}+\frac{1}{6}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x+1\right)+6=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2+6=3\left(5-y\right)$$

Tāpirihia te 2 ki te 6, ka 8.

$$2x+8=3\left(5-y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5-y.

$$2x+8=15-3y$$

Me tāpiri te 3y ki ngā taha e rua.

$$2x+8+3y=15$$

Tangohia te 8 mai i ngā taha e rua.

$$2x+3y=15-8$$

Tangohia te 8 i te 15, ka 7.

$$2x+3y=7$$

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.

$$2x-3y=1,2x+3y=7$$

Me tango 2x+3y=7 mai i 2x-3y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-3y-3y=1-7$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-3y=1-7$$

Tāpiri -3y ki te -3y.

$$-6y=1-7$$

Tāpiri 1 ki te -7.

$$-6y=-6$$

Whakawehea ngā taha e rua ki te -6.

$$y=1$$

Whakaurua te 1 mō y ki 2x+3y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3=7$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$