Whakaoti i te {l}{4x-ay-4a=0}{ax-4y+6a=0}

$$\left\{ \begin{array} { l } { 4 x - a y - 4 a = 0 } \\ { a x - 4 y + 6 a = 0 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16}$$

$$y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

$$a\neq -4\text{ and }a\neq 4$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

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

$$4x+\left(-a\right)y-4a=0$$

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

$$4x+\left(-a\right)y=4a$$

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

$$4x=ay+4a$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(ay+4a\right)$$

Whakareatia \frac{1}{4} ki te a\left(4+y\right).

$$x=\frac{a}{4}y+a$$

Whakakapia te a+\frac{ay}{4} mō te x ki tērā atu whārite, ax-4y+6a=0.

$$a\left(\frac{a}{4}y+a\right)-4y+6a=0$$

Whakareatia a ki te a+\frac{ay}{4}.

$$\frac{a^{2}}{4}y+a^{2}-4y+6a=0$$

Tāpiri \frac{a^{2}y}{4} ki te -4y.

$$\left(\frac{a^{2}}{4}-4\right)y+a^{2}+6a=0$$

Tāpiri a^{2} ki te 6a.

$$\left(\frac{a^{2}}{4}-4\right)y+a\left(a+6\right)=0$$

Me tango a\left(6+a\right) mai i ngā taha e rua o te whārite.

$$\left(\frac{a^{2}}{4}-4\right)y=-a\left(a+6\right)$$

Whakawehea ngā taha e rua ki te -4+\frac{a^{2}}{4}.

$$y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

Whakaurua te -\frac{4a\left(6+a\right)}{a^{2}-16} mō y ki x=\frac{a}{4}y+a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{a}{4}\left(-\frac{4a\left(a+6\right)}{a^{2}-16}\right)+a$$

Whakareatia \frac{a}{4} ki te -\frac{4a\left(6+a\right)}{a^{2}-16}.

$$x=-\frac{\left(a+6\right)a^{2}}{a^{2}-16}+a$$

Tāpiri a ki te -\frac{\left(6+a\right)a^{2}}{a^{2}-16}.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16},y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{4\left(-4\right)-\left(-a\right)a}&-\frac{-a}{4\left(-4\right)-\left(-a\right)a}\\-\frac{a}{4\left(-4\right)-\left(-a\right)a}&\frac{4}{4\left(-4\right)-\left(-a\right)a}\end{matrix}\right)\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{a^{2}-16}&\frac{a}{a^{2}-16}\\-\frac{a}{a^{2}-16}&\frac{4}{a^{2}-16}\end{matrix}\right)\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{4}{a^{2}-16}\right)\times 4a+\frac{a}{a^{2}-16}\left(-6a\right)\\\left(-\frac{a}{a^{2}-16}\right)\times 4a+\frac{4}{a^{2}-16}\left(-6a\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2a\left(3a+8\right)}{a^{2}-16}\\-\frac{4a\left(a+6\right)}{a^{2}-16}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16},y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

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

$$a\times 4x+a\left(-a\right)y+a\left(-4a\right)=0,4ax+4\left(-4\right)y+4\times 6a=0$$

Whakarūnātia.

$$4ax+\left(-a^{2}\right)y-4a^{2}=0,4ax-16y+24a=0$$

Me tango 4ax-16y+24a=0 mai i 4ax+\left(-a^{2}\right)y-4a^{2}=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4ax+\left(-4a\right)x+\left(-a^{2}\right)y+16y-4a^{2}-24a=0$$

Tāpiri 4ax ki te -4ax. Ka whakakore atu ngā kupu 4ax me -4ax, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-a^{2}\right)y+16y-4a^{2}-24a=0$$

Tāpiri -a^{2}y ki te 16y.

$$\left(16-a^{2}\right)y-4a^{2}-24a=0$$

Tāpiri -4a^{2} ki te -24a.

$$\left(16-a^{2}\right)y-4a\left(a+6\right)=0$$

Me tāpiri 4a\left(6+a\right) ki ngā taha e rua o te whārite.

$$\left(16-a^{2}\right)y=4a\left(a+6\right)$$

Whakawehea ngā taha e rua ki te -a^{2}+16.

$$y=\frac{4a\left(a+6\right)}{16-a^{2}}$$

Whakaurua te \frac{4a\left(6+a\right)}{16-a^{2}} mō y ki ax-4y+6a=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$ax-4\times \frac{4a\left(a+6\right)}{16-a^{2}}+6a=0$$

Whakareatia -4 ki te \frac{4a\left(6+a\right)}{16-a^{2}}.

$$ax-\frac{16a\left(a+6\right)}{16-a^{2}}+6a=0$$

Tāpiri -\frac{16a\left(6+a\right)}{16-a^{2}} ki te 6a.

$$ax-\frac{2\left(3a+8\right)a^{2}}{\left(4-a\right)\left(a+4\right)}=0$$

Me tāpiri \frac{2\left(8+3a\right)a^{2}}{\left(-a+4\right)\left(a+4\right)} ki ngā taha e rua o te whārite.

$$ax=\frac{2\left(3a+8\right)a^{2}}{\left(4-a\right)\left(a+4\right)}$$

Whakawehea ngā taha e rua ki te a.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{2a\left(3a+8\right)}{\left(4-a\right)\left(a+4\right)},y=\frac{4a\left(a+6\right)}{16-a^{2}}$$

Whakaoti mō x, y

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16}$$

$$y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

$$|a|\neq 4$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

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

$$4x+\left(-a\right)y-4a=0$$

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

$$4x+\left(-a\right)y=4a$$

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

$$4x=ay+4a$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(ay+4a\right)$$

Whakareatia \frac{1}{4} ki te a\left(4+y\right).

$$x=\frac{a}{4}y+a$$

Whakakapia te a+\frac{ay}{4} mō te x ki tērā atu whārite, ax-4y+6a=0.

$$a\left(\frac{a}{4}y+a\right)-4y+6a=0$$

Whakareatia a ki te a+\frac{ay}{4}.

$$\frac{a^{2}}{4}y+a^{2}-4y+6a=0$$

Tāpiri \frac{a^{2}y}{4} ki te -4y.

$$\left(\frac{a^{2}}{4}-4\right)y+a^{2}+6a=0$$

Tāpiri a^{2} ki te 6a.

$$\left(\frac{a^{2}}{4}-4\right)y+a\left(a+6\right)=0$$

Me tango a\left(6+a\right) mai i ngā taha e rua o te whārite.

$$\left(\frac{a^{2}}{4}-4\right)y=-a\left(a+6\right)$$

Whakawehea ngā taha e rua ki te -4+\frac{a^{2}}{4}.

$$y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

Whakaurua te -\frac{4a\left(6+a\right)}{a^{2}-16} mō y ki x=\frac{a}{4}y+a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{a}{4}\left(-\frac{4a\left(a+6\right)}{a^{2}-16}\right)+a$$

Whakareatia \frac{a}{4} ki te -\frac{4a\left(6+a\right)}{a^{2}-16}.

$$x=-\frac{\left(a+6\right)a^{2}}{a^{2}-16}+a$$

Tāpiri a ki te -\frac{\left(6+a\right)a^{2}}{a^{2}-16}.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16},y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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

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

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-a\\a&-4\end{matrix}\right))\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{4\left(-4\right)-\left(-a\right)a}&-\frac{-a}{4\left(-4\right)-\left(-a\right)a}\\-\frac{a}{4\left(-4\right)-\left(-a\right)a}&\frac{4}{4\left(-4\right)-\left(-a\right)a}\end{matrix}\right)\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{a^{2}-16}&\frac{a}{a^{2}-16}\\-\frac{a}{a^{2}-16}&\frac{4}{a^{2}-16}\end{matrix}\right)\left(\begin{matrix}4a\\-6a\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{4}{a^{2}-16}\right)\times 4a+\frac{a}{a^{2}-16}\left(-6a\right)\\\left(-\frac{a}{a^{2}-16}\right)\times 4a+\frac{4}{a^{2}-16}\left(-6a\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2a\left(3a+8\right)}{a^{2}-16}\\-\frac{4a\left(a+6\right)}{a^{2}-16}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{2a\left(3a+8\right)}{a^{2}-16},y=-\frac{4a\left(a+6\right)}{a^{2}-16}$$

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.

$$4x+\left(-a\right)y-4a=0,ax-4y+6a=0$$

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

$$a\times 4x+a\left(-a\right)y+a\left(-4a\right)=0,4ax+4\left(-4\right)y+4\times 6a=0$$

Whakarūnātia.

$$4ax+\left(-a^{2}\right)y-4a^{2}=0,4ax-16y+24a=0$$

Me tango 4ax-16y+24a=0 mai i 4ax+\left(-a^{2}\right)y-4a^{2}=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4ax+\left(-4a\right)x+\left(-a^{2}\right)y+16y-4a^{2}-24a=0$$

Tāpiri 4ax ki te -4ax. Ka whakakore atu ngā kupu 4ax me -4ax, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-a^{2}\right)y+16y-4a^{2}-24a=0$$

Tāpiri -a^{2}y ki te 16y.

$$\left(16-a^{2}\right)y-4a^{2}-24a=0$$

Tāpiri -4a^{2} ki te -24a.

$$\left(16-a^{2}\right)y-4a\left(a+6\right)=0$$

Me tāpiri 4a\left(6+a\right) ki ngā taha e rua o te whārite.

$$\left(16-a^{2}\right)y=4a\left(a+6\right)$$

Whakawehea ngā taha e rua ki te -a^{2}+16.

$$y=\frac{4a\left(a+6\right)}{16-a^{2}}$$

Whakaurua te \frac{4a\left(6+a\right)}{16-a^{2}} mō y ki ax-4y+6a=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$ax-4\times \frac{4a\left(a+6\right)}{16-a^{2}}+6a=0$$

Whakareatia -4 ki te \frac{4a\left(6+a\right)}{16-a^{2}}.

$$ax-\frac{16a\left(a+6\right)}{16-a^{2}}+6a=0$$

Tāpiri -\frac{16a\left(6+a\right)}{16-a^{2}} ki te 6a.

$$ax-\frac{2\left(3a+8\right)a^{2}}{\left(4-a\right)\left(a+4\right)}=0$$

Me tāpiri \frac{2\left(8+3a\right)a^{2}}{\left(-a+4\right)\left(a+4\right)} ki ngā taha e rua o te whārite.

$$ax=\frac{2\left(3a+8\right)a^{2}}{\left(4-a\right)\left(a+4\right)}$$

Whakawehea ngā taha e rua ki te a.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{2a\left(3a+8\right)}{\left(4-a\right)\left(a+4\right)},y=\frac{4a\left(a+6\right)}{16-a^{2}}$$