Whakaoti i te {l}{3x+y=2}{5x-y=8}

$$\left\{ \begin{array} { l } { 3 x + y = 2 } \\ { 5 x - y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

$$y = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

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.

$$3x+y=2,5x-y=8$$

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

$$3x+y=2$$

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

$$3x=-y+2$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-y+2\right)$$

Whakareatia \frac{1}{3} ki te -y+2.

$$x=-\frac{1}{3}y+\frac{2}{3}$$

Whakakapia te \frac{-y+2}{3} mō te x ki tērā atu whārite, 5x-y=8.

$$5\left(-\frac{1}{3}y+\frac{2}{3}\right)-y=8$$

Whakareatia 5 ki te \frac{-y+2}{3}.

$$-\frac{5}{3}y+\frac{10}{3}-y=8$$

Tāpiri -\frac{5y}{3} ki te -y.

$$-\frac{8}{3}y+\frac{10}{3}=8$$

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

$$-\frac{8}{3}y=\frac{14}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{7}{4}$$

Whakaurua te -\frac{7}{4} mō y ki x=-\frac{1}{3}y+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\left(-\frac{7}{4}\right)+\frac{2}{3}$$

Whakareatia -\frac{1}{3} ki te -\frac{7}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{7}{12}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te \frac{7}{12} 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.

$$x=\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{4},y=-\frac{7}{4}$$

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.

$$3x+y=2,5x-y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\5&-1\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4}\\-\frac{7}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{4},y=-\frac{7}{4}$$

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.

$$3x+y=2,5x-y=8$$

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5y=5\times 2,3\times 5x+3\left(-1\right)y=3\times 8$$

Whakarūnātia.

$$15x+5y=10,15x-3y=24$$

Me tango 15x-3y=24 mai i 15x+5y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+5y+3y=10-24$$

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.

$$5y+3y=10-24$$

Tāpiri 5y ki te 3y.

$$8y=10-24$$

Tāpiri 10 ki te -24.

$$8y=-14$$

Whakawehea ngā taha e rua ki te 8.

$$y=-\frac{7}{4}$$

Whakaurua te -\frac{7}{4} mō y ki 5x-y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\left(-\frac{7}{4}\right)=8$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

$$5x=\frac{25}{4}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{4},y=-\frac{7}{4}$$