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

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

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }y=2\text{, }&\text{unconditionally}\\x=\frac{4\left(y-2\right)}{3}\text{, }y\in \mathrm{C}\text{, }&a=\frac{3}{2}\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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.

$$4y-3x=8,-2y+ax=-4$$

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

$$4y-3x=8$$

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

$$4y=3x+8$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{4}\left(3x+8\right)$$

Whakareatia \frac{1}{4} ki te 3x+8.

$$y=\frac{3}{4}x+2$$

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

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

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

$$-\frac{3}{2}x-4+ax=-4$$

Tāpiri -\frac{3x}{2} ki te ax.

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

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

$$\left(a-\frac{3}{2}\right)x=0$$

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

$$x=0$$

Whakaurua te 0 mō x ki y=\frac{3}{4}x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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

$$4y-3x=8,-2y+ax=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{a}{2\left(2a-3\right)}&\frac{3}{2\left(2a-3\right)}\\\frac{1}{2a-3}&\frac{2}{2a-3}\end{matrix}\right)\left(\begin{matrix}8\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{a}{2\left(2a-3\right)}\times 8+\frac{3}{2\left(2a-3\right)}\left(-4\right)\\\frac{1}{2a-3}\times 8+\frac{2}{2a-3}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=2,x=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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.

$$4y-3x=8,-2y+ax=-4$$

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

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

Whakarūnātia.

$$-8y+6x=-16,-8y+4ax=-16$$

Me tango -8y+4ax=-16 mai i -8y+6x=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8y+8y+6x+\left(-4a\right)x=-16+16$$

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.

$$6x+\left(-4a\right)x=-16+16$$

Tāpiri 6x ki te -4ax.

$$\left(6-4a\right)x=-16+16$$

Tāpiri -16 ki te 16.

$$\left(6-4a\right)x=0$$

Whakawehea ngā taha e rua ki te 6-4a.

$$x=0$$

Whakaurua te 0 mō x ki -2y+ax=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=0$$

Whakaoti mō x, y

$$\left\{\begin{matrix}\\x=0\text{, }y=2\text{, }&\text{unconditionally}\\x=\frac{4\left(y-2\right)}{3}\text{, }y\in \mathrm{R}\text{, }&a=\frac{3}{2}\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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.

$$4y-3x=8,-2y+ax=-4$$

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

$$4y-3x=8$$

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

$$4y=3x+8$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{4}\left(3x+8\right)$$

Whakareatia \frac{1}{4} ki te 3x+8.

$$y=\frac{3}{4}x+2$$

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

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

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

$$-\frac{3}{2}x-4+ax=-4$$

Tāpiri -\frac{3x}{2} ki te ax.

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

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

$$\left(a-\frac{3}{2}\right)x=0$$

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

$$x=0$$

Whakaurua te 0 mō x ki y=\frac{3}{4}x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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

$$4y-3x=8,-2y+ax=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{a}{2\left(2a-3\right)}&\frac{3}{2\left(2a-3\right)}\\\frac{1}{2a-3}&\frac{2}{2a-3}\end{matrix}\right)\left(\begin{matrix}8\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{a}{2\left(2a-3\right)}\times 8+\frac{3}{2\left(2a-3\right)}\left(-4\right)\\\frac{1}{2a-3}\times 8+\frac{2}{2a-3}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=2,x=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2y mai i ngā taha e rua.

$$ax+4-2y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$ax-2y=-4$$

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.

$$4y-3x=8,-2y+ax=-4$$

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

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

Whakarūnātia.

$$-8y+6x=-16,-8y+4ax=-16$$

Me tango -8y+4ax=-16 mai i -8y+6x=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8y+8y+6x+\left(-4a\right)x=-16+16$$

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.

$$6x+\left(-4a\right)x=-16+16$$

Tāpiri 6x ki te -4ax.

$$\left(6-4a\right)x=-16+16$$

Tāpiri -16 ki te 16.

$$\left(6-4a\right)x=0$$

Whakawehea ngā taha e rua ki te 6-4a.

$$x=0$$

Whakaurua te 0 mō x ki -2y+ax=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=0$$