Whakaoti i te {l}{x/2+y/3=10}{x+y=30}

$$\left. \begin{array} { l } { \frac { x } { 2 } + \frac { y } { 3 } = 10 } \\ { x + y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=30$$

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.

$$\frac{1}{2}x+\frac{1}{3}y=10,x+y=30$$

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

$$\frac{1}{2}x+\frac{1}{3}y=10$$

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

$$\frac{1}{2}x=-\frac{1}{3}y+10$$

Me whakarea ngā taha e rua ki te 2.

$$x=2\left(-\frac{1}{3}y+10\right)$$

Whakareatia 2 ki te -\frac{y}{3}+10.

$$x=-\frac{2}{3}y+20$$

Whakakapia te -\frac{2y}{3}+20 mō te x ki tērā atu whārite, x+y=30.

$$-\frac{2}{3}y+20+y=30$$

Tāpiri -\frac{2y}{3} ki te y.

$$\frac{1}{3}y+20=30$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$\frac{1}{3}y=10$$

Me whakarea ngā taha e rua ki te 3.

$$y=30$$

Whakaurua te 30 mō y ki x=-\frac{2}{3}y+20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times 30+20$$

Whakareatia -\frac{2}{3} ki te 30.

$$x=-20+20$$

Tāpiri 20 ki te -20.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=30$$

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.

$$\frac{1}{2}x+\frac{1}{3}y=10,x+y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\30\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right))\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&\frac{1}{3}\\1&1\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{\frac{1}{2}-\frac{1}{3}}&-\frac{\frac{1}{3}}{\frac{1}{2}-\frac{1}{3}}\\-\frac{1}{\frac{1}{2}-\frac{1}{3}}&\frac{\frac{1}{2}}{\frac{1}{2}-\frac{1}{3}}\end{matrix}\right)\left(\begin{matrix}10\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6&-2\\-6&3\end{matrix}\right)\left(\begin{matrix}10\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\times 10-2\times 30\\-6\times 10+3\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=30$$

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.

$$\frac{1}{2}x+\frac{1}{3}y=10,x+y=30$$

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

$$\frac{1}{2}x+\frac{1}{3}y=10,\frac{1}{2}x+\frac{1}{2}y=\frac{1}{2}\times 30$$

Whakarūnātia.

$$\frac{1}{2}x+\frac{1}{3}y=10,\frac{1}{2}x+\frac{1}{2}y=15$$

Me tango \frac{1}{2}x+\frac{1}{2}y=15 mai i \frac{1}{2}x+\frac{1}{3}y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{2}x-\frac{1}{2}x+\frac{1}{3}y-\frac{1}{2}y=10-15$$

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

$$\frac{1}{3}y-\frac{1}{2}y=10-15$$

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

$$-\frac{1}{6}y=10-15$$

Tāpiri 10 ki te -15.

$$-\frac{1}{6}y=-5$$

Me whakarea ngā taha e rua ki te -6.

$$y=30$$

Whakaurua te 30 mō y ki x+y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+30=30$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=30$$