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

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

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.

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

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

$$5x+y=8$$

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

$$5x=-y+8$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-y+8\right)$$

Whakareatia \frac{1}{5} ki te -y+8.

$$x=-\frac{1}{5}y+\frac{8}{5}$$

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

$$3\left(-\frac{1}{5}y+\frac{8}{5}\right)-y=8$$

Whakareatia 3 ki te \frac{-y+8}{5}.

$$-\frac{3}{5}y+\frac{24}{5}-y=8$$

Tāpiri -\frac{3y}{5} ki te -y.

$$-\frac{8}{5}y+\frac{24}{5}=8$$

Me tango \frac{24}{5} mai i ngā taha e rua o te whārite.

$$-\frac{8}{5}y=\frac{16}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{1}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\left(-2\right)+\frac{8}{5}$$

Whakareatia -\frac{1}{5} ki te -2.

$$x=\frac{2+8}{5}$$

Tāpiri \frac{8}{5} ki te \frac{2}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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.

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

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3y=3\times 8,5\times 3x+5\left(-1\right)y=5\times 8$$

Whakarūnātia.

$$15x+3y=24,15x-5y=40$$

Me tango 15x-5y=40 mai i 15x+3y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+3y+5y=24-40$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+5y=24-40$$

Tāpiri 3y ki te 5y.

$$8y=24-40$$

Tāpiri 24 ki te -40.

$$8y=-16$$

Whakawehea ngā taha e rua ki te 8.

$$y=-2$$

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.

$$3x-\left(-2\right)=8$$

Me tango 2 mai i 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.

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