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

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

Whakaoti mō x, y

$$x=8$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 5.

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

Whakareatia te 3 ki te 5, ka 15.

$$2x-y=15$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 3.

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

Whakareatia te 3 ki te 3, ka 9.

$$x+y=9$$

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.

$$2x-y=15,x+y=9$$

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

$$2x-y=15$$

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

$$2x=y+15$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y+15\right)$$

Whakareatia \frac{1}{2} ki te y+15.

$$x=\frac{1}{2}y+\frac{15}{2}$$

Whakakapia te \frac{15+y}{2} mō te x ki tērā atu whārite, x+y=9.

$$\frac{1}{2}y+\frac{15}{2}+y=9$$

Tāpiri \frac{y}{2} ki te y.

$$\frac{3}{2}y+\frac{15}{2}=9$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$\frac{3}{2}y=\frac{3}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{1}{2}y+\frac{15}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1+15}{2}$$

Tāpiri \frac{15}{2} ki te \frac{1}{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.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 5.

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

Whakareatia te 3 ki te 5, ka 15.

$$2x-y=15$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 3.

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

Whakareatia te 3 ki te 3, ka 9.

$$x+y=9$$

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

$$2x-y=15,x+y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-1\\1&1\end{matrix}\right))\left(\begin{matrix}2&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\1&1\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\1&1\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\1&1\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-1\right)}&-\frac{-1}{2-\left(-1\right)}\\-\frac{1}{2-\left(-1\right)}&\frac{2}{2-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}15\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\-\frac{1}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}15\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 15+\frac{1}{3}\times 9\\-\frac{1}{3}\times 15+\frac{2}{3}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 5.

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

Whakareatia te 3 ki te 5, ka 15.

$$2x-y=15$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 3.

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

Whakareatia te 3 ki te 3, ka 9.

$$x+y=9$$

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.

$$2x-y=15,x+y=9$$

Kia ōrite ai a 2x 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 2.

$$2x-y=15,2x+2y=2\times 9$$

Whakarūnātia.

$$2x-y=15,2x+2y=18$$

Me tango 2x+2y=18 mai i 2x-y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-y-2y=15-18$$

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.

$$-y-2y=15-18$$

Tāpiri -y ki te -2y.

$$-3y=15-18$$

Tāpiri 15 ki te -18.

$$-3y=-3$$

Whakawehea ngā taha e rua ki te -3.

$$y=1$$

Whakaurua te 1 mō y ki x+y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+1=9$$

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

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=1$$