Whakaoti i te {c}{27/5+4/5*y=-4*x+3/5}{8*x=-8-3*y}

$$\left.\begin{array} { c } { \frac { 27 } { 5 } + \frac { 4 } { 5 } \cdot y = \frac { - 4 \cdot x + 3 } { 5 } } \\ { 8 \cdot x = - 8 - 3 \cdot y } \end{array} \right\}$$

Whakaoti mō y, x

$$x=2$$

$$y=-8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$27+4y=-4x+3$$

Me tāpiri te 4x ki ngā taha e rua.

$$27+4y+4x=3$$

Tangohia te 27 mai i ngā taha e rua.

$$4y+4x=3-27$$

Tangohia te 27 i te 3, ka -24.

$$4y+4x=-24$$

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$8x+3y=-8$$

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.

$$4y+4x=-24,3y+8x=-8$$

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

$$4y+4x=-24$$

Me tango 4x mai i ngā taha e rua o te whārite.

$$4y=-4x-24$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{4}\left(-4x-24\right)$$

Whakareatia \frac{1}{4} ki te -4x-24.

$$y=-x-6$$

Whakakapia te -x-6 mō te y ki tērā atu whārite, 3y+8x=-8.

$$3\left(-x-6\right)+8x=-8$$

Whakareatia 3 ki te -x-6.

$$-3x-18+8x=-8$$

Tāpiri -3x ki te 8x.

$$5x-18=-8$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$5x=10$$

Whakawehea ngā taha e rua ki te 5.

$$x=2$$

Whakaurua te 2 mō x ki y=-x-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2-6$$

Tāpiri -6 ki te -2.

$$y=-8$$

Kua oti te pūnaha te whakatau.

$$y=-8,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$27+4y=-4x+3$$

Me tāpiri te 4x ki ngā taha e rua.

$$27+4y+4x=3$$

Tangohia te 27 mai i ngā taha e rua.

$$4y+4x=3-27$$

Tangohia te 27 i te 3, ka -24.

$$4y+4x=-24$$

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$8x+3y=-8$$

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

$$4y+4x=-24,3y+8x=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&4\\3&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&4\\3&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&4\\3&8\end{matrix}\right))\left(\begin{matrix}4&4\\3&8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\3&8\end{matrix}\right))\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\3&8\end{matrix}\right))\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&4\\3&8\end{matrix}\right))\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{4\times 8-4\times 3}&-\frac{4}{4\times 8-4\times 3}\\-\frac{3}{4\times 8-4\times 3}&\frac{4}{4\times 8-4\times 3}\end{matrix}\right)\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\-\frac{3}{20}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-24\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\left(-24\right)-\frac{1}{5}\left(-8\right)\\-\frac{3}{20}\left(-24\right)+\frac{1}{5}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-8,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$27+4y=-4x+3$$

Me tāpiri te 4x ki ngā taha e rua.

$$27+4y+4x=3$$

Tangohia te 27 mai i ngā taha e rua.

$$4y+4x=3-27$$

Tangohia te 27 i te 3, ka -24.

$$4y+4x=-24$$

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$8x+3y=-8$$

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.

$$4y+4x=-24,3y+8x=-8$$

Kia ōrite ai a 4y me 3y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$3\times 4y+3\times 4x=3\left(-24\right),4\times 3y+4\times 8x=4\left(-8\right)$$

Whakarūnātia.

$$12y+12x=-72,12y+32x=-32$$

Me tango 12y+32x=-32 mai i 12y+12x=-72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12y-12y+12x-32x=-72+32$$

Tāpiri 12y ki te -12y. Ka whakakore atu ngā kupu 12y me -12y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12x-32x=-72+32$$

Tāpiri 12x ki te -32x.

$$-20x=-72+32$$

Tāpiri -72 ki te 32.

$$-20x=-40$$

Whakawehea ngā taha e rua ki te -20.

$$x=2$$

Whakaurua te 2 mō x ki 3y+8x=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3y+8\times 2=-8$$

Whakareatia 8 ki te 2.

$$3y+16=-8$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$3y=-24$$

Whakawehea ngā taha e rua ki te 3.

$$y=-8$$

Kua oti te pūnaha te whakatau.

$$y=-8,x=2$$