Whakaoti i te {l}{3x-5y=1}{10y-4x=6}

$$\left\{ \begin{array} { l } { 3 x - 5 y = 1 } \\ { 10 y - 4 x = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

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.

$$3x-5y=1,-4x+10y=6$$

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

$$3x-5y=1$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te \frac{5y+1}{3} mō te x ki tērā atu whārite, -4x+10y=6.

$$-4\left(\frac{5}{3}y+\frac{1}{3}\right)+10y=6$$

Whakareatia -4 ki te \frac{5y+1}{3}.

$$-\frac{20}{3}y-\frac{4}{3}+10y=6$$

Tāpiri -\frac{20y}{3} ki te 10y.

$$\frac{10}{3}y-\frac{4}{3}=6$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$\frac{10}{3}y=\frac{22}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{10}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki x=\frac{5}{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.

$$x=\frac{5}{3}\times \frac{11}{5}+\frac{1}{3}$$

Whakareatia \frac{5}{3} ki te \frac{11}{5} 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.

$$x=\frac{11+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{11}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=\frac{11}{5}$$

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.

$$3x-5y=1,-4x+10y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\-4&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right))\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&10\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{3\times 10-\left(-5\left(-4\right)\right)}&-\frac{-5}{3\times 10-\left(-5\left(-4\right)\right)}\\-\frac{-4}{3\times 10-\left(-5\left(-4\right)\right)}&\frac{3}{3\times 10-\left(-5\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&\frac{1}{2}\\\frac{2}{5}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1+\frac{1}{2}\times 6\\\frac{2}{5}+\frac{3}{10}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\\frac{11}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=\frac{11}{5}$$

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.

$$3x-5y=1,-4x+10y=6$$

Kia ōrite ai a 3x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-4\times 3x-4\left(-5\right)y=-4,3\left(-4\right)x+3\times 10y=3\times 6$$

Whakarūnātia.

$$-12x+20y=-4,-12x+30y=18$$

Me tango -12x+30y=18 mai i -12x+20y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+20y-30y=-4-18$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-30y=-4-18$$

Tāpiri 20y ki te -30y.

$$-10y=-4-18$$

Tāpiri -4 ki te -18.

$$-10y=-22$$

Whakawehea ngā taha e rua ki te -10.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki -4x+10y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+10\times \frac{11}{5}=6$$

Whakareatia 10 ki te \frac{11}{5}.

$$-4x+22=6$$

Me tango 22 mai i ngā taha e rua o te whārite.

$$-4x=-16$$

Whakawehea ngā taha e rua ki te -4.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=\frac{11}{5}$$