Whakaoti i te 3x-y=5,5x+3y=-1

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

Whakaoti mō x, y

$$x=1$$

$$y=-2$$

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.

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

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

$$3x-y=5$$

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

$$3x=y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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

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

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

$$\frac{5}{3}y+\frac{25}{3}+3y=-1$$

Tāpiri \frac{5y}{3} ki te 3y.

$$\frac{14}{3}y+\frac{25}{3}=-1$$

Me tango \frac{25}{3} mai i ngā taha e rua o te whārite.

$$\frac{14}{3}y=-\frac{28}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{3}, 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}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

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

Tāpiri \frac{5}{3} ki te -\frac{2}{3} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}&\frac{1}{14}\\-\frac{5}{14}&\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}5\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}\times 5+\frac{1}{14}\left(-1\right)\\-\frac{5}{14}\times 5+\frac{3}{14}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-2$$

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.

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

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.

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

Whakarūnātia.

$$15x-5y=25,15x+9y=-3$$

Me tango 15x+9y=-3 mai i 15x-5y=25 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-5y-9y=25+3$$

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.

$$-5y-9y=25+3$$

Tāpiri -5y ki te -9y.

$$-14y=25+3$$

Tāpiri 25 ki te 3.

$$-14y=28$$

Whakawehea ngā taha e rua ki te -14.

$$y=-2$$

Whakaurua te -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.

$$5x+3\left(-2\right)=-1$$

Whakareatia 3 ki te -2.

$$5x-6=-1$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$