Whakaoti i te {l}{2y=3x-6}{5x+4y=8}

$$\left\{ \begin{array} { l } { 2 y = 3 x - 6 } \\ { 5 x + 4 y = 8 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{20}{11} = 1\frac{9}{11} \approx 1.818181818$$

$$y=-\frac{3}{11}\approx -0.272727273$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

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.

$$2y-3x=-6,4y+5x=8$$

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

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

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

$$2y=3x-6$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1}{2}\left(3x-6\right)$$

Whakareatia \frac{1}{2} ki te -6+3x.

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

Whakakapia te \frac{3x}{2}-3 mō te y ki tērā atu whārite, 4y+5x=8.

$$4\left(\frac{3}{2}x-3\right)+5x=8$$

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

$$6x-12+5x=8$$

Tāpiri 6x ki te 5x.

$$11x-12=8$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$11x=20$$

Whakawehea ngā taha e rua ki te 11.

$$x=\frac{20}{11}$$

Whakaurua te \frac{20}{11} mō x ki y=\frac{3}{2}x-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{3}{2}\times \frac{20}{11}-3$$

Whakareatia \frac{3}{2} ki te \frac{20}{11} 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.

$$y=\frac{30}{11}-3$$

Tāpiri -3 ki te \frac{30}{11}.

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

Kua oti te pūnaha te whakatau.

$$y=-\frac{3}{11},x=\frac{20}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

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

$$2y-3x=-6,4y+5x=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\4&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-6\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\4&5\end{matrix}\right))\left(\begin{matrix}2&-3\\4&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-6\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-\left(-3\times 4\right)}&-\frac{-3}{2\times 5-\left(-3\times 4\right)}\\-\frac{4}{2\times 5-\left(-3\times 4\right)}&\frac{2}{2\times 5-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-6\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{22}&\frac{3}{22}\\-\frac{2}{11}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}-6\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{22}\left(-6\right)+\frac{3}{22}\times 8\\-\frac{2}{11}\left(-6\right)+\frac{1}{11}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{11}\\\frac{20}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{3}{11},x=\frac{20}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

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

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

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.

$$2y-3x=-6,4y+5x=8$$

Kia ōrite ai a 2y me 4y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2y+4\left(-3\right)x=4\left(-6\right),2\times 4y+2\times 5x=2\times 8$$

Whakarūnātia.

$$8y-12x=-24,8y+10x=16$$

Me tango 8y+10x=16 mai i 8y-12x=-24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8y-8y-12x-10x=-24-16$$

Tāpiri 8y ki te -8y. Ka whakakore atu ngā kupu 8y me -8y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12x-10x=-24-16$$

Tāpiri -12x ki te -10x.

$$-22x=-24-16$$

Tāpiri -24 ki te -16.

$$-22x=-40$$

Whakawehea ngā taha e rua ki te -22.

$$x=\frac{20}{11}$$

Whakaurua te \frac{20}{11} mō x ki 4y+5x=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4y+5\times \frac{20}{11}=8$$

Whakareatia 5 ki te \frac{20}{11}.

$$4y+\frac{100}{11}=8$$

Me tango \frac{100}{11} mai i ngā taha e rua o te whārite.

$$4y=-\frac{12}{11}$$

Whakawehea ngā taha e rua ki te 4.

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

Kua oti te pūnaha te whakatau.

$$y=-\frac{3}{11},x=\frac{20}{11}$$