Whakaoti i te {l}{x_{1}+x_{2}=97}{frac{x_{2}}{x_{1}}=2}

$$\left\{ \begin{array} { l } { x _ { 1 } + x _ { 2 } = 97 } \\ { \frac { x _ { 2 } } { x _ { 1 } } = 2 } \end{array} \right.$$

Whakaoti mō x_1, x_2

$$x_{1} = \frac{97}{3} = 32\frac{1}{3} \approx 32.333333333$$

$$x_{2} = \frac{194}{3} = 64\frac{2}{3} \approx 64.666666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x_{1} ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x_{1}.

$$x_{2}=2x_{1}$$

Tangohia te 2x_{1} mai i ngā taha e rua.

$$x_{2}-2x_{1}=0$$

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.

$$x_{1}+x_{2}=97,-2x_{1}+x_{2}=0$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x_{1} mā te wehe i te x_{1} i te taha mauī o te tohu ōrite.

$$x_{1}+x_{2}=97$$

Me tango x_{2} mai i ngā taha e rua o te whārite.

$$x_{1}=-x_{2}+97$$

Whakakapia te -x_{2}+97 mō te x_{1} ki tērā atu whārite, -2x_{1}+x_{2}=0.

$$-2\left(-x_{2}+97\right)+x_{2}=0$$

Whakareatia -2 ki te -x_{2}+97.

$$2x_{2}-194+x_{2}=0$$

Tāpiri 2x_{2} ki te x_{2}.

$$3x_{2}-194=0$$

Me tāpiri 194 ki ngā taha e rua o te whārite.

$$3x_{2}=194$$

Whakawehea ngā taha e rua ki te 3.

$$x_{2}=\frac{194}{3}$$

Whakaurua te \frac{194}{3} mō x_{2} ki x_{1}=-x_{2}+97. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x_{1} hāngai tonu.

$$x_{1}=-\frac{194}{3}+97$$

Tāpiri 97 ki te -\frac{194}{3}.

$$x_{1}=\frac{97}{3}$$

Kua oti te pūnaha te whakatau.

$$x_{1}=\frac{97}{3},x_{2}=\frac{194}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x_{1} ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x_{1}.

$$x_{2}=2x_{1}$$

Tangohia te 2x_{1} mai i ngā taha e rua.

$$x_{2}-2x_{1}=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x_{1}+x_{2}=97,-2x_{1}+x_{2}=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-2&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}97\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}1&1\\-2&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}97\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}97\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}97\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-2\right)}&-\frac{1}{1-\left(-2\right)}\\-\frac{-2}{1-\left(-2\right)}&\frac{1}{1-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}97\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}97\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 97\\\frac{2}{3}\times 97\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{97}{3}\\\frac{194}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x_{1} me x_{2}.

$$x_{1}=\frac{97}{3},x_{2}=\frac{194}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x_{1} ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x_{1}.

$$x_{2}=2x_{1}$$

Tangohia te 2x_{1} mai i ngā taha e rua.

$$x_{2}-2x_{1}=0$$

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.

$$x_{1}+x_{2}=97,-2x_{1}+x_{2}=0$$

Me tango -2x_{1}+x_{2}=0 mai i x_{1}+x_{2}=97 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x_{1}+2x_{1}+x_{2}-x_{2}=97$$

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

$$x_{1}+2x_{1}=97$$

Tāpiri x_{1} ki te 2x_{1}.

$$3x_{1}=97$$

Whakawehea ngā taha e rua ki te 3.

$$x_{1}=\frac{97}{3}$$

Whakaurua te \frac{97}{3} mō x_{1} ki -2x_{1}+x_{2}=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x_{2} hāngai tonu.

$$-2\times \frac{97}{3}+x_{2}=0$$

Whakareatia -2 ki te \frac{97}{3}.

$$-\frac{194}{3}+x_{2}=0$$

Me tāpiri \frac{194}{3} ki ngā taha e rua o te whārite.

$$x_{2}=\frac{194}{3}$$

Kua oti te pūnaha te whakatau.

$$x_{1}=\frac{97}{3},x_{2}=\frac{194}{3}$$