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

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

Whakaoti mō x, y

$$x=1$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

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

Hei kimi i te tauaro o y+3, kimihia te tauaro o ia taurangi.

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

Tangohia te 6x mai i ngā taha e rua.

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

Pahekotia te 2x me -6x, ka -4x.

$$-4x-y-3=2y+2$$

Tangohia te 2y mai i ngā taha e rua.

$$-4x-y-3-2y=2$$

Pahekotia te -y me -2y, ka -3y.

$$-4x-3y-3=2$$

Me tāpiri te 3 ki ngā taha e rua.

$$-4x-3y=2+3$$

Tāpirihia te 2 ki te 3, ka 5.

$$-4x-3y=5$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$5x+y=4x-2$$

Tangohia te 4x mai i ngā taha e rua.

$$5x+y-4x=-2$$

Pahekotia te 5x me -4x, ka x.

$$x+y=-2$$

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.

$$-4x-3y=5,x+y=-2$$

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

$$-4x-3y=5$$

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

$$-4x=3y+5$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(3y+5\right)$$

Whakareatia -\frac{1}{4} ki te 3y+5.

$$x=-\frac{3}{4}y-\frac{5}{4}$$

Whakakapia te \frac{-3y-5}{4} mō te x ki tērā atu whārite, x+y=-2.

$$-\frac{3}{4}y-\frac{5}{4}+y=-2$$

Tāpiri -\frac{3y}{4} ki te y.

$$\frac{1}{4}y-\frac{5}{4}=-2$$

Me tāpiri \frac{5}{4} ki ngā taha e rua o te whārite.

$$\frac{1}{4}y=-\frac{3}{4}$$

Me whakarea ngā taha e rua ki te 4.

$$y=-3$$

Whakaurua te -3 mō y ki x=-\frac{3}{4}y-\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-3\right)-\frac{5}{4}$$

Whakareatia -\frac{3}{4} ki te -3.

$$x=\frac{9-5}{4}$$

Tāpiri -\frac{5}{4} ki te \frac{9}{4} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

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

Hei kimi i te tauaro o y+3, kimihia te tauaro o ia taurangi.

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

Tangohia te 6x mai i ngā taha e rua.

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

Pahekotia te 2x me -6x, ka -4x.

$$-4x-y-3=2y+2$$

Tangohia te 2y mai i ngā taha e rua.

$$-4x-y-3-2y=2$$

Pahekotia te -y me -2y, ka -3y.

$$-4x-3y-3=2$$

Me tāpiri te 3 ki ngā taha e rua.

$$-4x-3y=2+3$$

Tāpirihia te 2 ki te 3, ka 5.

$$-4x-3y=5$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$5x+y=4x-2$$

Tangohia te 4x mai i ngā taha e rua.

$$5x+y-4x=-2$$

Pahekotia te 5x me -4x, ka x.

$$x+y=-2$$

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

$$-4x-3y=5,x+y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&-3\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-4-\left(-3\right)}&-\frac{-3}{-4-\left(-3\right)}\\-\frac{1}{-4-\left(-3\right)}&-\frac{4}{-4-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-3\\1&4\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5-3\left(-2\right)\\5+4\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

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

Hei kimi i te tauaro o y+3, kimihia te tauaro o ia taurangi.

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

Tangohia te 6x mai i ngā taha e rua.

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

Pahekotia te 2x me -6x, ka -4x.

$$-4x-y-3=2y+2$$

Tangohia te 2y mai i ngā taha e rua.

$$-4x-y-3-2y=2$$

Pahekotia te -y me -2y, ka -3y.

$$-4x-3y-3=2$$

Me tāpiri te 3 ki ngā taha e rua.

$$-4x-3y=2+3$$

Tāpirihia te 2 ki te 3, ka 5.

$$-4x-3y=5$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$5x+y=4x-2$$

Tangohia te 4x mai i ngā taha e rua.

$$5x+y-4x=-2$$

Pahekotia te 5x me -4x, ka x.

$$x+y=-2$$

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.

$$-4x-3y=5,x+y=-2$$

Kia ōrite ai a -4x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -4.

$$-4x-3y=5,-4x-4y=-4\left(-2\right)$$

Whakarūnātia.

$$-4x-3y=5,-4x-4y=8$$

Me tango -4x-4y=8 mai i -4x-3y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-3y+4y=5-8$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y+4y=5-8$$

Tāpiri -3y ki te 4y.

$$y=5-8$$

Tāpiri 5 ki te -8.

$$y=-3$$

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

$$x-3=-2$$

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

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-3$$