Whakaoti i te {c}{2x+3y-10=0}{4x=3y+20}

$$\left. \begin{array} { c } { 2 x + 3 y - 10 = 0 } \\ { 4 x = 3 y + 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x+3y=10$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=20$$

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.

$$2x+3y=10,4x-3y=20$$

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

$$2x+3y=10$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+10\right)$$

Whakareatia \frac{1}{2} ki te -3y+10.

$$x=-\frac{3}{2}y+5$$

Whakakapia te -\frac{3y}{2}+5 mō te x ki tērā atu whārite, 4x-3y=20.

$$4\left(-\frac{3}{2}y+5\right)-3y=20$$

Whakareatia 4 ki te -\frac{3y}{2}+5.

$$-6y+20-3y=20$$

Tāpiri -6y ki te -3y.

$$-9y+20=20$$

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

$$-9y=0$$

Whakawehea ngā taha e rua ki te -9.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{3}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x+3y=10$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=20$$

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

$$2x+3y=10,4x-3y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\20\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&3\\4&-3\end{matrix}\right))\left(\begin{matrix}2&3\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-3\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-3\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&-3\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-3\times 4}&-\frac{3}{2\left(-3\right)-3\times 4}\\-\frac{4}{2\left(-3\right)-3\times 4}&\frac{2}{2\left(-3\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}10\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}10\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 10+\frac{1}{6}\times 20\\\frac{2}{9}\times 10-\frac{1}{9}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x+3y=10$$

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=20$$

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.

$$2x+3y=10,4x-3y=20$$

Kia ōrite ai a 2x me 4x, 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.

$$4\times 2x+4\times 3y=4\times 10,2\times 4x+2\left(-3\right)y=2\times 20$$

Whakarūnātia.

$$8x+12y=40,8x-6y=40$$

Me tango 8x-6y=40 mai i 8x+12y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+12y+6y=40-40$$

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

$$12y+6y=40-40$$

Tāpiri 12y ki te 6y.

$$18y=40-40$$

Tāpiri 40 ki te -40.

$$18y=0$$

Whakawehea ngā taha e rua ki te 18.

$$y=0$$

Whakaurua te 0 mō y ki 4x-3y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x=20$$

Whakawehea ngā taha e rua ki te 4.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$