Whakaoti i te {l}{x-5y=8}{3x+y=-8}

$$\left. \begin{array} { l } { x - 5 y = 8 } \\ { 3 x + y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-2$$

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.

$$x-5y=8,3x+y=-8$$

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

$$x-5y=8$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$x=5y+8$$

Whakakapia te 5y+8 mō te x ki tērā atu whārite, 3x+y=-8.

$$3\left(5y+8\right)+y=-8$$

Whakareatia 3 ki te 5y+8.

$$15y+24+y=-8$$

Tāpiri 15y ki te y.

$$16y+24=-8$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$16y=-32$$

Whakawehea ngā taha e rua ki te 16.

$$y=-2$$

Whakaurua te -2 mō y ki x=5y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\left(-2\right)+8$$

Whakareatia 5 ki te -2.

$$x=-10+8$$

Tāpiri 8 ki te -10.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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

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.

$$x-5y=8,3x+y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-5\\3&1\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\3&1\end{matrix}\right))\left(\begin{matrix}8\\-8\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{16}&\frac{5}{16}\\-\frac{3}{16}&\frac{1}{16}\end{matrix}\right)\left(\begin{matrix}8\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{16}\times 8+\frac{5}{16}\left(-8\right)\\-\frac{3}{16}\times 8+\frac{1}{16}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x-5y=8,3x+y=-8$$

Kia ōrite ai a x me 3x, 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 1.

$$3x+3\left(-5\right)y=3\times 8,3x+y=-8$$

Whakarūnātia.

$$3x-15y=24,3x+y=-8$$

Me tango 3x+y=-8 mai i 3x-15y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-15y-y=24+8$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-y=24+8$$

Tāpiri -15y ki te -y.

$$-16y=24+8$$

Tāpiri 24 ki te 8.

$$-16y=32$$

Whakawehea ngā taha e rua ki te -16.

$$y=-2$$

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

$$3x-2=-8$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$3x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

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