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

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

Whakaoti mō x, y

$$x=3$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakareatia te 5 ki te 2, ka 10.

$$5y-1=3x+10$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-1-3x=10$$

Me tāpiri te 1 ki ngā taha e rua.

$$5y-3x=10+1$$

Tāpirihia te 10 ki te 1, ka 11.

$$5y-3x=11$$

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.

$$3x-y=5,-3x+5y=11$$

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

$$3x-y=5$$

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

$$3x=y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{1}{3}y+\frac{5}{3}$$

Whakakapia te \frac{5+y}{3} mō te x ki tērā atu whārite, -3x+5y=11.

$$-3\left(\frac{1}{3}y+\frac{5}{3}\right)+5y=11$$

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

$$-y-5+5y=11$$

Tāpiri -y ki te 5y.

$$4y-5=11$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$4y=16$$

Whakawehea ngā taha e rua ki te 4.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{1}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 4+\frac{5}{3}$$

Whakareatia \frac{1}{3} ki te 4.

$$x=\frac{4+5}{3}$$

Tāpiri \frac{5}{3} ki te \frac{4}{3} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakareatia te 5 ki te 2, ka 10.

$$5y-1=3x+10$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-1-3x=10$$

Me tāpiri te 1 ki ngā taha e rua.

$$5y-3x=10+1$$

Tāpirihia te 10 ki te 1, ka 11.

$$5y-3x=11$$

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

$$3x-y=5,-3x+5y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\11\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right))\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-3&5\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-\left(-\left(-3\right)\right)}&-\frac{-1}{3\times 5-\left(-\left(-3\right)\right)}\\-\frac{-3}{3\times 5-\left(-\left(-3\right)\right)}&\frac{3}{3\times 5-\left(-\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}&\frac{1}{12}\\\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}5\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}\times 5+\frac{1}{12}\times 11\\\frac{1}{4}\times 5+\frac{1}{4}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakareatia te 5 ki te 2, ka 10.

$$5y-1=3x+10$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-1-3x=10$$

Me tāpiri te 1 ki ngā taha e rua.

$$5y-3x=10+1$$

Tāpirihia te 10 ki te 1, ka 11.

$$5y-3x=11$$

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.

$$3x-y=5,-3x+5y=11$$

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

$$-3\times 3x-3\left(-1\right)y=-3\times 5,3\left(-3\right)x+3\times 5y=3\times 11$$

Whakarūnātia.

$$-9x+3y=-15,-9x+15y=33$$

Me tango -9x+15y=33 mai i -9x+3y=-15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+3y-15y=-15-33$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-15y=-15-33$$

Tāpiri 3y ki te -15y.

$$-12y=-15-33$$

Tāpiri -15 ki te -33.

$$-12y=-48$$

Whakawehea ngā taha e rua ki te -12.

$$y=4$$

Whakaurua te 4 mō y ki -3x+5y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+5\times 4=11$$

Whakareatia 5 ki te 4.

$$-3x+20=11$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-3x=-9$$

Whakawehea ngā taha e rua ki te -3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$