Whakaoti i te {l}{2bx+ay=2ab}{bx-ay=4ab}

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

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}\\x=2a\text{, }y=-2b\text{, }&\text{unconditionally}\\x=0\text{, }y\in \mathrm{C}\text{, }&a=0\\x\in \mathrm{C}\text{, }y=0\text{, }&b=0\\x\in \mathrm{C}\text{, }y\in \mathrm{C}\text{, }&b=0\text{ and }a=0\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

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

$$2bx+ay=2ab$$

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

$$2bx=\left(-a\right)y+2ab$$

Whakawehea ngā taha e rua ki te 2b.

$$x=\frac{1}{2b}\left(\left(-a\right)y+2ab\right)$$

Whakareatia \frac{1}{2b} ki te a\left(-y+2b\right).

$$x=\left(-\frac{a}{2b}\right)y+a$$

Whakakapia te a-\frac{ay}{2b} mō te x ki tērā atu whārite, bx+\left(-a\right)y=4ab.

$$b\left(\left(-\frac{a}{2b}\right)y+a\right)+\left(-a\right)y=4ab$$

Whakareatia b ki te a-\frac{ay}{2b}.

$$\left(-\frac{a}{2}\right)y+ab+\left(-a\right)y=4ab$$

Tāpiri -\frac{ay}{2} ki te -ay.

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

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

$$\left(-\frac{3a}{2}\right)y=3ab$$

Whakawehea ngā taha e rua ki te -\frac{3a}{2}.

$$y=-2b$$

Whakaurua te -2b mō y ki x=\left(-\frac{a}{2b}\right)y+a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{a}{2b}\right)\left(-2b\right)+a$$

Whakareatia -\frac{a}{2b} ki te -2b.

$$x=a+a$$

Tāpiri a ki te a.

$$x=2a$$

Kua oti te pūnaha te whakatau.

$$x=2a,y=-2b$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{a}{2b\left(-a\right)-ab}&-\frac{a}{2b\left(-a\right)-ab}\\-\frac{b}{2b\left(-a\right)-ab}&\frac{2b}{2b\left(-a\right)-ab}\end{matrix}\right)\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2a\\-2b\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2a,y=-2b$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

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

$$b\times 2bx+bay=b\times 2ab,2bbx+2b\left(-a\right)y=2b\times 4ab$$

Whakarūnātia.

$$2b^{2}x+aby=2ab^{2},2b^{2}x+\left(-2ab\right)y=8ab^{2}$$

Me tango 2b^{2}x+\left(-2ab\right)y=8ab^{2} mai i 2b^{2}x+aby=2ab^{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2b^{2}x+\left(-2b^{2}\right)x+aby+2aby=2ab^{2}-8ab^{2}$$

Tāpiri 2b^{2}x ki te -2b^{2}x. Ka whakakore atu ngā kupu 2b^{2}x me -2b^{2}x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$aby+2aby=2ab^{2}-8ab^{2}$$

Tāpiri bay ki te 2bay.

$$3aby=2ab^{2}-8ab^{2}$$

Tāpiri 2ab^{2} ki te -8ab^{2}.

$$3aby=-6ab^{2}$$

Whakawehea ngā taha e rua ki te 3ba.

$$y=-2b$$

Whakaurua te -2b mō y ki bx+\left(-a\right)y=4ab. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$bx+\left(-a\right)\left(-2b\right)=4ab$$

Whakareatia -a ki te -2b.

$$bx+2ab=4ab$$

Me tango 2ba mai i ngā taha e rua o te whārite.

$$bx=2ab$$

Whakawehea ngā taha e rua ki te b.

$$x=2a$$

Kua oti te pūnaha te whakatau.

$$x=2a,y=-2b$$

Whakaoti mō x, y

$$\left\{\begin{matrix}\\x=2a\text{, }y=-2b\text{, }&\text{unconditionally}\\x=0\text{, }y\in \mathrm{R}\text{, }&a=0\\x\in \mathrm{R}\text{, }y=0\text{, }&b=0\\x\in \mathrm{R}\text{, }y\in \mathrm{R}\text{, }&b=0\text{ and }a=0\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

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

$$2bx+ay=2ab$$

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

$$2bx=\left(-a\right)y+2ab$$

Whakawehea ngā taha e rua ki te 2b.

$$x=\frac{1}{2b}\left(\left(-a\right)y+2ab\right)$$

Whakareatia \frac{1}{2b} ki te a\left(-y+2b\right).

$$x=\left(-\frac{a}{2b}\right)y+a$$

Whakakapia te a-\frac{ay}{2b} mō te x ki tērā atu whārite, bx+\left(-a\right)y=4ab.

$$b\left(\left(-\frac{a}{2b}\right)y+a\right)+\left(-a\right)y=4ab$$

Whakareatia b ki te a-\frac{ay}{2b}.

$$\left(-\frac{a}{2}\right)y+ab+\left(-a\right)y=4ab$$

Tāpiri -\frac{ay}{2} ki te -ay.

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

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

$$\left(-\frac{3a}{2}\right)y=3ab$$

Whakawehea ngā taha e rua ki te -\frac{3a}{2}.

$$y=-2b$$

Whakaurua te -2b mō y ki x=\left(-\frac{a}{2b}\right)y+a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{a}{2b}\right)\left(-2b\right)+a$$

Whakareatia -\frac{a}{2b} ki te -2b.

$$x=a+a$$

Tāpiri a ki te a.

$$x=2a$$

Kua oti te pūnaha te whakatau.

$$x=2a,y=-2b$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2b&a\\b&-a\end{matrix}\right))\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{a}{2b\left(-a\right)-ab}&-\frac{a}{2b\left(-a\right)-ab}\\-\frac{b}{2b\left(-a\right)-ab}&\frac{2b}{2b\left(-a\right)-ab}\end{matrix}\right)\left(\begin{matrix}2ab\\4ab\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2a\\-2b\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2a,y=-2b$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$2bx+ay=2ab,bx+\left(-a\right)y=4ab$$

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

$$b\times 2bx+bay=b\times 2ab,2bbx+2b\left(-a\right)y=2b\times 4ab$$

Whakarūnātia.

$$2b^{2}x+aby=2ab^{2},2b^{2}x+\left(-2ab\right)y=8ab^{2}$$

Me tango 2b^{2}x+\left(-2ab\right)y=8ab^{2} mai i 2b^{2}x+aby=2ab^{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2b^{2}x+\left(-2b^{2}\right)x+aby+2aby=2ab^{2}-8ab^{2}$$

Tāpiri 2b^{2}x ki te -2b^{2}x. Ka whakakore atu ngā kupu 2b^{2}x me -2b^{2}x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$aby+2aby=2ab^{2}-8ab^{2}$$

Tāpiri bay ki te 2bay.

$$3aby=2ab^{2}-8ab^{2}$$

Tāpiri 2ab^{2} ki te -8ab^{2}.

$$3aby=-6ab^{2}$$

Whakawehea ngā taha e rua ki te 3ba.

$$y=-2b$$

Whakaurua te -2b mō y ki bx+\left(-a\right)y=4ab. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$bx+\left(-a\right)\left(-2b\right)=4ab$$

Whakareatia -a ki te -2b.

$$bx+2ab=4ab$$

Me tango 2ba mai i ngā taha e rua o te whārite.

$$bx=2ab$$

Whakawehea ngā taha e rua ki te b.

$$x=2a$$

Kua oti te pūnaha te whakatau.

$$x=2a,y=-2b$$