Whakaoti i te {l}{3x+y=3}{5x-y=15}

$$\left\{ \begin{array} { l } { 3 x + y = 3 } \\ { 5 x - y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

$$y = -\frac{15}{4} = -3\frac{3}{4} = -3.75$$

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.

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

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

$$3x+y=3$$

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

$$3x=-y+3$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te -\frac{y}{3}+1 mō te x ki tērā atu whārite, 5x-y=15.

$$5\left(-\frac{1}{3}y+1\right)-y=15$$

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

$$-\frac{5}{3}y+5-y=15$$

Tāpiri -\frac{5y}{3} ki te -y.

$$-\frac{8}{3}y+5=15$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

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.

$$y=-\frac{15}{4}$$

Whakaurua te -\frac{15}{4} mō y ki x=-\frac{1}{3}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\left(-\frac{15}{4}\right)+1$$

Whakareatia -\frac{1}{3} ki te -\frac{15}{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.

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

Tāpiri 1 ki te \frac{5}{4}.

$$x=\frac{9}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{4},y=-\frac{15}{4}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\5&-1\end{matrix}\right))\left(\begin{matrix}3&1\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\5&-1\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\5&-1\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\5&-1\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-5}&-\frac{1}{3\left(-1\right)-5}\\-\frac{5}{3\left(-1\right)-5}&\frac{3}{3\left(-1\right)-5}\end{matrix}\right)\left(\begin{matrix}3\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&\frac{1}{8}\\\frac{5}{8}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}3\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 3+\frac{1}{8}\times 15\\\frac{5}{8}\times 3-\frac{3}{8}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{4}\\-\frac{15}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{4},y=-\frac{15}{4}$$

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.

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

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.

$$5\times 3x+5y=5\times 3,3\times 5x+3\left(-1\right)y=3\times 15$$

Whakarūnātia.

$$15x+5y=15,15x-3y=45$$

Me tango 15x-3y=45 mai i 15x+5y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+5y+3y=15-45$$

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.

$$5y+3y=15-45$$

Tāpiri 5y ki te 3y.

$$8y=15-45$$

Tāpiri 15 ki te -45.

$$8y=-30$$

Whakawehea ngā taha e rua ki te 8.

$$y=-\frac{15}{4}$$

Whakaurua te -\frac{15}{4} mō y ki 5x-y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\left(-\frac{15}{4}\right)=15$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$5x=\frac{45}{4}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{9}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{4},y=-\frac{15}{4}$$