Whakaoti i te {l}{2y=3x-4}{2y=x+1}

$$\left. \begin{array} { l } { 2 y = 3 x - 4 } \\ { 2 y = x + 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$y = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

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

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

$$2y-x=1$$

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.

$$2y-3x=-4,2y-x=1$$

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

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

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

$$2y=3x-4$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1}{2}\left(3x-4\right)$$

Whakareatia \frac{1}{2} ki te 3x-4.

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

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

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

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

$$3x-4-x=1$$

Tāpiri 3x ki te -x.

$$2x-4=1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x=5$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō x ki y=\frac{3}{2}x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{3}{2}\times \frac{5}{2}-2$$

Whakareatia \frac{3}{2} ki te \frac{5}{2} 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.

$$y=\frac{15}{4}-2$$

Tāpiri -2 ki te \frac{15}{4}.

$$y=\frac{7}{4}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{7}{4},x=\frac{5}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

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

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

$$2y-x=1$$

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

$$2y-3x=-4,2y-x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\2&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4}\\\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{7}{4},x=\frac{5}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

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

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

$$2y-x=1$$

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.

$$2y-3x=-4,2y-x=1$$

Me tango 2y-x=1 mai i 2y-3x=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-3x+x=-4-1$$

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

$$-3x+x=-4-1$$

Tāpiri -3x ki te x.

$$-2x=-4-1$$

Tāpiri -4 ki te -1.

$$-2x=-5$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō x ki 2y-x=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

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

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$2y=\frac{7}{2}$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{7}{4}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{7}{4},x=\frac{5}{2}$$