Whakaoti i te {l}{y=x+3/2+3}{y=2x+10}

$$\left. \begin{array} { l } { y = \frac { x + 3 } { 2 } + 3 } \\ { y = 2 x + 10 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{11}{3} = -3\frac{2}{3} \approx -3.666666667$$

$$y = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawehea ia wā o x+3 ki te 2, kia riro ko \frac{1}{2}x+\frac{3}{2}.

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

Tāpirihia te \frac{3}{2} ki te 3, ka \frac{9}{2}.

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

Whakakapia te \frac{9+x}{2} mō te y ki tērā atu whārite, y-2x=10.

$$\frac{1}{2}x+\frac{9}{2}-2x=10$$

Tāpiri \frac{x}{2} ki te -2x.

$$-\frac{3}{2}x+\frac{9}{2}=10$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$-\frac{3}{2}x=\frac{11}{2}$$

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.

$$x=-\frac{11}{3}$$

Whakaurua te -\frac{11}{3} mō x ki y=\frac{1}{2}x+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{2}\left(-\frac{11}{3}\right)+\frac{9}{2}$$

Whakareatia \frac{1}{2} ki te -\frac{11}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=-\frac{11}{6}+\frac{9}{2}$$

Tāpiri \frac{9}{2} ki te -\frac{11}{6} 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.

$$y=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{8}{3},x=-\frac{11}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawehea ia wā o x+3 ki te 2, kia riro ko \frac{1}{2}x+\frac{3}{2}.

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

Tāpirihia te \frac{3}{2} ki te 3, ka \frac{9}{2}.

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

Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$y-\frac{1}{2}x=\frac{9}{2}$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=10$$

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

$$y-\frac{1}{2}x=\frac{9}{2},y-2x=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-\frac{1}{2}\right)}&-\frac{-\frac{1}{2}}{-2-\left(-\frac{1}{2}\right)}\\-\frac{1}{-2-\left(-\frac{1}{2}\right)}&\frac{1}{-2-\left(-\frac{1}{2}\right)}\end{matrix}\right)\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}&-\frac{1}{3}\\\frac{2}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\times \frac{9}{2}-\frac{1}{3}\times 10\\\frac{2}{3}\times \frac{9}{2}-\frac{2}{3}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\\-\frac{11}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{8}{3},x=-\frac{11}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawehea ia wā o x+3 ki te 2, kia riro ko \frac{1}{2}x+\frac{3}{2}.

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

Tāpirihia te \frac{3}{2} ki te 3, ka \frac{9}{2}.

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

Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$y-\frac{1}{2}x=\frac{9}{2}$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=10$$

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.

$$y-\frac{1}{2}x=\frac{9}{2},y-2x=10$$

Me tango y-2x=10 mai i y-\frac{1}{2}x=\frac{9}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-\frac{1}{2}x+2x=\frac{9}{2}-10$$

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

$$-\frac{1}{2}x+2x=\frac{9}{2}-10$$

Tāpiri -\frac{x}{2} ki te 2x.

$$\frac{3}{2}x=\frac{9}{2}-10$$

Tāpiri \frac{9}{2} ki te -10.

$$\frac{3}{2}x=-\frac{11}{2}$$

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.

$$x=-\frac{11}{3}$$

Whakaurua te -\frac{11}{3} mō x ki y-2x=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-2\left(-\frac{11}{3}\right)=10$$

Whakareatia -2 ki te -\frac{11}{3}.

$$y+\frac{22}{3}=10$$

Me tango \frac{22}{3} mai i ngā taha e rua o te whārite.

$$y=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{8}{3},x=-\frac{11}{3}$$