Whakaoti i te bx+cy=a+b,ax[1/a-b-1/a+b]+cy[1/b-a-1/b+a]=frac{2a}{a+b}

$$b x + c y = a + b , a x [ \frac { 1 } { a - b } - \frac { 1 } { a + b } ] + c y [ \frac { 1 } { b - a } - \frac { 1 } { b + a } ] = \frac { 2 a } { a + b }$$

Whakaoti mō x, y (complex solution)

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

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

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

$$bx+cy=a+b$$

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

$$bx=\left(-c\right)y+a+b$$

Whakawehea ngā taha e rua ki te b.

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

Whakareatia \frac{1}{b} ki te -cy+a+b.

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

Whakakapia te \frac{-cy+a+b}{b} mō te x ki tērā atu whārite, \left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}.

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

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

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

Tāpiri -\frac{2acy}{\left(a-b\right)\left(a+b\right)} ki te \frac{2cay}{\left(b-a\right)\left(b+a\right)}.

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

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

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

Whakawehea ngā taha e rua ki te \frac{4ca}{\left(b-a\right)\left(a+b\right)}.

$$y=\frac{b}{c}$$

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

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

Whakareatia -\frac{c}{b} ki te \frac{b}{c}.

$$x=-1+\frac{a+b}{b}$$

Tāpiri \frac{a+b}{b} ki te -1.

$$x=\frac{a}{b}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{a}{b},y=\frac{b}{c}$$

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{a}{b},y=\frac{b}{c}$$

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

Kia ōrite ai a bx me \frac{2abx}{\left(a-b\right)\left(a+b\right)}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te a\left(\left(a-b\right)^{-1}-\left(a+b\right)^{-1}\right) me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te b.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)abx+\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)acy=\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)a\left(a+b\right),b\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+b\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=b\times \frac{2a}{a+b}$$

Whakarūnātia.

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

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

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

Tāpiri \frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)} ki te -\frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)}. Ka whakakore atu ngā kupu \frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)} me -\frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri \frac{2abcy}{\left(a-b\right)\left(a+b\right)} ki te -\frac{2bcay}{\left(b-a\right)\left(b+a\right)}.

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

Tāpiri \frac{2ab}{a-b} ki te -\frac{2ba}{a+b}.

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

Whakawehea ngā taha e rua ki te \frac{4bca}{\left(a-b\right)\left(a+b\right)}.

$$y=\frac{b}{c}$$

Whakaurua te \frac{b}{c} mō y ki \left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)c\times \frac{b}{c}=\frac{2a}{a+b}$$

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

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

Me tango \frac{2ab}{\left(b-a\right)\left(b+a\right)} mai i ngā taha e rua o te whārite.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax=-\frac{2a^{2}}{\left(b-a\right)\left(a+b\right)}$$

Whakawehea ngā taha e rua ki te a\left(\left(a-b\right)^{-1}-\left(a+b\right)^{-1}\right).

$$x=\frac{a}{b}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{a}{b},y=\frac{b}{c}$$

Whakaoti mō x, y

$$\left\{\begin{matrix}x=\frac{a}{b}\text{, }y=\frac{b}{c}\text{, }&c\neq 0\text{ and }b\neq 0\text{ and }|b|\neq |a|\\x=\frac{b-cy}{b}\text{, }y\in \mathrm{R}\text{, }&a=0\text{ and }b\neq 0\end{matrix}\right.$$

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

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

$$bx+cy=a+b$$

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

$$bx=\left(-c\right)y+a+b$$

Whakawehea ngā taha e rua ki te b.

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

Whakareatia \frac{1}{b} ki te -cy+a+b.

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

Whakakapia te \frac{-cy+a+b}{b} mō te x ki tērā atu whārite, \left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}.

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

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

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

Tāpiri -\frac{2acy}{\left(a-b\right)\left(a+b\right)} ki te \frac{2cay}{\left(b-a\right)\left(b+a\right)}.

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

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

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

Whakawehea ngā taha e rua ki te \frac{4ca}{\left(b-a\right)\left(a+b\right)}.

$$y=\frac{b}{c}$$

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

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

Whakareatia -\frac{c}{b} ki te \frac{b}{c}.

$$x=-1+\frac{a+b}{b}$$

Tāpiri \frac{a+b}{b} ki te -1\text{, }|b|\neq |a|.

$$x=\frac{a}{b}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{a}{b},y=\frac{b}{c}$$

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{a}{b},y=\frac{b}{c}$$

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.

$$bx+cy=a+b,\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}$$

Kia ōrite ai a bx me \frac{2abx}{\left(a-b\right)\left(a+b\right)}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te a\left(\left(a-b\right)^{-1}-\left(a+b\right)^{-1}\right) me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te b.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)abx+\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)acy=\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)a\left(a+b\right),b\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+b\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=b\times \frac{2a}{a+b}$$

Whakarūnātia.

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

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

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

Tāpiri \frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)} ki te -\frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)}. Ka whakakore atu ngā kupu \frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)} me -\frac{2ab^{2}x}{\left(a-b\right)\left(a+b\right)}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri \frac{2abcy}{\left(a-b\right)\left(a+b\right)} ki te -\frac{2bcay}{\left(b-a\right)\left(b+a\right)}.

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

Tāpiri \frac{2ab}{a-b} ki te -\frac{2ba}{a+b}.

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

Whakawehea ngā taha e rua ki te \frac{4bca}{\left(a-b\right)\left(a+b\right)}.

$$y=\frac{b}{c}$$

Whakaurua te \frac{b}{c} mō y ki \left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)cy=\frac{2a}{a+b}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax+\left(-\frac{1}{a+b}+\frac{1}{b-a}\right)c\times \frac{b}{c}=\frac{2a}{a+b}$$

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

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

Me tango \frac{2ab}{\left(b-a\right)\left(b+a\right)} mai i ngā taha e rua o te whārite.

$$\left(-\frac{1}{a+b}+\frac{1}{a-b}\right)ax=-\frac{2a^{2}}{\left(b-a\right)\left(a+b\right)}$$

Whakawehea ngā taha e rua ki te a\left(\left(a-b\right)^{-1}-\left(a+b\right)^{-1}\right).

$$x=\frac{a}{b}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{a}{b},y=\frac{b}{c}$$