Whakaoti i te {l}{10x+10y=9}{5x-2y=1}

$$\left\{ \begin{array} { l } { 10 x + 10 y = 9 } \\ { 5 x - 2 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{2}{5}=0.4$$

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

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.

$$10x+10y=9,5x-2y=1$$

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

$$10x+10y=9$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$10x=-10y+9$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-10y+9\right)$$

Whakareatia \frac{1}{10} ki te -10y+9.

$$x=-y+\frac{9}{10}$$

Whakakapia te -y+\frac{9}{10} mō te x ki tērā atu whārite, 5x-2y=1.

$$5\left(-y+\frac{9}{10}\right)-2y=1$$

Whakareatia 5 ki te -y+\frac{9}{10}.

$$-5y+\frac{9}{2}-2y=1$$

Tāpiri -5y ki te -2y.

$$-7y+\frac{9}{2}=1$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$-7y=-\frac{7}{2}$$

Whakawehea ngā taha e rua ki te -7.

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

Whakaurua te \frac{1}{2} mō y ki x=-y+\frac{9}{10}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}+\frac{9}{10}$$

Tāpiri \frac{9}{10} ki te -\frac{1}{2} 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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{5},y=\frac{1}{2}$$

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.

$$10x+10y=9,5x-2y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&10\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&10\\5&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&10\\5&-2\end{matrix}\right))\left(\begin{matrix}10&10\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&10\\5&-2\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&10\\5&-2\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&10\\5&-2\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{10\left(-2\right)-10\times 5}&-\frac{10}{10\left(-2\right)-10\times 5}\\-\frac{5}{10\left(-2\right)-10\times 5}&\frac{10}{10\left(-2\right)-10\times 5}\end{matrix}\right)\left(\begin{matrix}9\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{35}&\frac{1}{7}\\\frac{1}{14}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}9\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{35}\times 9+\frac{1}{7}\\\frac{1}{14}\times 9-\frac{1}{7}\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2}{5},y=\frac{1}{2}$$

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.

$$10x+10y=9,5x-2y=1$$

Kia ōrite ai a 10x 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 10.

$$5\times 10x+5\times 10y=5\times 9,10\times 5x+10\left(-2\right)y=10$$

Whakarūnātia.

$$50x+50y=45,50x-20y=10$$

Me tango 50x-20y=10 mai i 50x+50y=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x+50y+20y=45-10$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$50y+20y=45-10$$

Tāpiri 50y ki te 20y.

$$70y=45-10$$

Tāpiri 45 ki te -10.

$$70y=35$$

Whakawehea ngā taha e rua ki te 70.

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

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

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

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

$$5x-1=1$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$5x=2$$

Whakawehea ngā taha e rua ki te 5.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{5},y=\frac{1}{2}$$