Whakaoti i te {l}{5x-2y=4}{x/2+y/3=2}

$$\left\{ \begin{array} { l } { 5 x - 2 y = 4 } \\ { \frac { x } { 2 } + \frac { y } { 3 } = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

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.

$$5x-2y=4,\frac{1}{2}x+\frac{1}{3}y=2$$

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

$$5x-2y=4$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$5x=2y+4$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+4\right)$$

Whakareatia \frac{1}{5} ki te 4+2y.

$$x=\frac{2}{5}y+\frac{4}{5}$$

Whakakapia te \frac{4+2y}{5} mō te x ki tērā atu whārite, \frac{1}{2}x+\frac{1}{3}y=2.

$$\frac{1}{2}\left(\frac{2}{5}y+\frac{4}{5}\right)+\frac{1}{3}y=2$$

Whakareatia \frac{1}{2} ki te \frac{4+2y}{5}.

$$\frac{1}{5}y+\frac{2}{5}+\frac{1}{3}y=2$$

Tāpiri \frac{y}{5} ki te \frac{y}{3}.

$$\frac{8}{15}y+\frac{2}{5}=2$$

Me tango \frac{2}{5} mai i ngā taha e rua o te whārite.

$$\frac{8}{15}y=\frac{8}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{15}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{2}{5}y+\frac{4}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times 3+\frac{4}{5}$$

Whakareatia \frac{2}{5} ki te 3.

$$x=\frac{6+4}{5}$$

Tāpiri \frac{4}{5} ki te \frac{6}{5} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

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.

$$5x-2y=4,\frac{1}{2}x+\frac{1}{3}y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\\frac{1}{2}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}4\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{3}}{5\times \frac{1}{3}-\left(-2\times \frac{1}{2}\right)}&-\frac{-2}{5\times \frac{1}{3}-\left(-2\times \frac{1}{2}\right)}\\-\frac{\frac{1}{2}}{5\times \frac{1}{3}-\left(-2\times \frac{1}{2}\right)}&\frac{5}{5\times \frac{1}{3}-\left(-2\times \frac{1}{2}\right)}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&\frac{3}{4}\\-\frac{3}{16}&\frac{15}{8}\end{matrix}\right)\left(\begin{matrix}4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 4+\frac{3}{4}\times 2\\-\frac{3}{16}\times 4+\frac{15}{8}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

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.

$$5x-2y=4,\frac{1}{2}x+\frac{1}{3}y=2$$

Kia ōrite ai a 5x me \frac{x}{2}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{2} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$\frac{1}{2}\times 5x+\frac{1}{2}\left(-2\right)y=\frac{1}{2}\times 4,5\times \frac{1}{2}x+5\times \frac{1}{3}y=5\times 2$$

Whakarūnātia.

$$\frac{5}{2}x-y=2,\frac{5}{2}x+\frac{5}{3}y=10$$

Me tango \frac{5}{2}x+\frac{5}{3}y=10 mai i \frac{5}{2}x-y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{5}{2}x-\frac{5}{2}x-y-\frac{5}{3}y=2-10$$

Tāpiri \frac{5x}{2} ki te -\frac{5x}{2}. Ka whakakore atu ngā kupu \frac{5x}{2} me -\frac{5x}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-y-\frac{5}{3}y=2-10$$

Tāpiri -y ki te -\frac{5y}{3}.

$$-\frac{8}{3}y=2-10$$

Tāpiri 2 ki te -10.

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

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.

$$y=3$$

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

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

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

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

Me tango 1 mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$