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

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

Whakaoti mō x, y

$$x = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

$$y=\frac{1}{3}\approx 0.333333333$$

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.

$$x+2y=5,x-y=4$$

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

$$x+2y=5$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+5$$

Whakakapia te -2y+5 mō te x ki tērā atu whārite, x-y=4.

$$-2y+5-y=4$$

Tāpiri -2y ki te -y.

$$-3y+5=4$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-3y=-1$$

Whakawehea ngā taha e rua ki te -3.

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

Whakaurua te \frac{1}{3} mō y ki x=-2y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{1}{3}+5$$

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

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

Tāpiri 5 ki te -\frac{2}{3}.

$$x=\frac{13}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{3},y=\frac{1}{3}$$

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.

$$x+2y=5,x-y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{3}\\\frac{1}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{13}{3},y=\frac{1}{3}$$

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.

$$x+2y=5,x-y=4$$

Me tango x-y=4 mai i x+2y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+2y+y=5-4$$

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

$$2y+y=5-4$$

Tāpiri 2y ki te y.

$$3y=5-4$$

Tāpiri 5 ki te -4.

$$3y=1$$

Whakawehea ngā taha e rua ki te 3.

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

Whakaurua te \frac{1}{3} mō y ki x-y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{1}{3}=4$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$x=\frac{13}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{3},y=\frac{1}{3}$$