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

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

Whakaoti mō x, y

$$x = \frac{9}{8} = 1\frac{1}{8} = 1.125$$

$$y = \frac{15}{8} = 1\frac{7}{8} = 1.875$$

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.

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

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

$$x+y=3$$

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

$$x=-y+3$$

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

$$-\left(-y+3\right)+y=\frac{3}{4}$$

Whakareatia -1 ki te -y+3.

$$y-3+y=\frac{3}{4}$$

Tāpiri y ki te y.

$$2y-3=\frac{3}{4}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$2y=\frac{15}{4}$$

Whakawehea ngā taha e rua ki te 2.

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

Whakaurua te \frac{15}{8} mō y ki x=-y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{15}{8}+3$$

Tāpiri 3 ki te -\frac{15}{8}.

$$x=\frac{9}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{8},y=\frac{15}{8}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{8}\\\frac{15}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{8},y=\frac{15}{8}$$

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.

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

Me tango -x+y=\frac{3}{4} mai i x+y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+x+y-y=3-\frac{3}{4}$$

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

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

Tāpiri x ki te x.

$$2x=3-\frac{3}{4}$$

Tāpiri 3 ki te -\frac{3}{4}.

$$2x=\frac{9}{4}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{9}{8}$$

Whakaurua te \frac{9}{8} mō x ki -x+y=\frac{3}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-\frac{9}{8}+y=\frac{3}{4}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{8},y=\frac{15}{8}$$