Whakaoti i te {l}{3x-y=1}{5x-3y=1}

$$\left\{ \begin{array} { l } { 3 x - y = 1 } \\ { 5 x - 3 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=\frac{1}{2}=0.5$$

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.

$$3x-y=1,5x-3y=1$$

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

$$3x-y=1$$

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

$$3x=y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{1+y}{3} mō te x ki tērā atu whārite, 5x-3y=1.

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

Whakareatia 5 ki te \frac{1+y}{3}.

$$\frac{5}{3}y+\frac{5}{3}-3y=1$$

Tāpiri \frac{5y}{3} ki te -3y.

$$-\frac{4}{3}y+\frac{5}{3}=1$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

$$-\frac{4}{3}y=-\frac{2}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{1}{2}$$

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

$$x=\frac{1}{3}\times \frac{1}{2}+\frac{1}{3}$$

Whakareatia \frac{1}{3} ki te \frac{1}{2} 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.

$$x=\frac{1}{6}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{1}{6} 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.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{1}{2}$$

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.

$$3x-y=1,5x-3y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&-\frac{1}{4}\\\frac{5}{4}&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3-1}{4}\\\frac{5-3}{4}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\\frac{1}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=\frac{1}{2}$$

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.

$$3x-y=1,5x-3y=1$$

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.

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

Whakarūnātia.

$$15x-5y=5,15x-9y=3$$

Me tango 15x-9y=3 mai i 15x-5y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-5y+9y=5-3$$

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.

$$-5y+9y=5-3$$

Tāpiri -5y ki te 9y.

$$4y=5-3$$

Tāpiri 5 ki te -3.

$$4y=2$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō y ki 5x-3y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-3\times \frac{1}{2}=1$$

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

$$5x-\frac{3}{2}=1$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{1}{2}$$