Whakaoti i te {l}{3.4x=x+y/2}{4y=x+y/2}

$$\left. \begin{array} { l } { 3.4 x = \frac { x + y } { 2 } } \\ { 4 y = \frac { x + y } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

$$6.8x=x+y$$

Tangohia te x mai i ngā taha e rua.

$$6.8x-x=y$$

Pahekotia te 6.8x me -x, ka 5.8x.

$$5.8x=y$$

Whakawehea ngā taha e rua o te whārite ki te 5.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{5}{29}y$$

Whakakapia te \frac{5y}{29} mō te x ki tērā atu whārite, -x+7y=0.

$$-\frac{5}{29}y+7y=0$$

Tāpiri -\frac{5y}{29} ki te 7y.

$$\frac{198}{29}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{198}{29}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{5}{29}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

$$6.8x=x+y$$

Tangohia te x mai i ngā taha e rua.

$$6.8x-x=y$$

Pahekotia te 6.8x me -x, ka 5.8x.

$$5.8x=y$$

Tangohia te y mai i ngā taha e rua.

$$5.8x-y=0$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$8y=x+y$$

Tangohia te x mai i ngā taha e rua.

$$8y-x=y$$

Tangohia te y mai i ngā taha e rua.

$$8y-x-y=0$$

Pahekotia te 8y me -y, ka 7y.

$$7y-x=0$$

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

$$5.8x-y=0,-x+7y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right))\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5.8&-1\\-1&7\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5.8\times 7-\left(-\left(-1\right)\right)}&-\frac{-1}{5.8\times 7-\left(-\left(-1\right)\right)}\\-\frac{-1}{5.8\times 7-\left(-\left(-1\right)\right)}&\frac{5.8}{5.8\times 7-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{198}&\frac{5}{198}\\\frac{5}{198}&\frac{29}{198}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 2.

$$6.8x=x+y$$

Tangohia te x mai i ngā taha e rua.

$$6.8x-x=y$$

Pahekotia te 6.8x me -x, ka 5.8x.

$$5.8x=y$$

Tangohia te y mai i ngā taha e rua.

$$5.8x-y=0$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$8y=x+y$$

Tangohia te x mai i ngā taha e rua.

$$8y-x=y$$

Tangohia te y mai i ngā taha e rua.

$$8y-x-y=0$$

Pahekotia te 8y me -y, ka 7y.

$$7y-x=0$$

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.

$$5.8x-y=0,-x+7y=0$$

Kia ōrite ai a \frac{29x}{5} 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 5.8.

$$-5.8x-\left(-y\right)=0,5.8\left(-1\right)x+5.8\times 7y=0$$

Whakarūnātia.

$$-5.8x+y=0,-5.8x+40.6y=0$$

Me tango -5.8x+40.6y=0 mai i -5.8x+y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5.8x+5.8x+y-40.6y=0$$

Tāpiri -\frac{29x}{5} ki te \frac{29x}{5}. Ka whakakore atu ngā kupu -\frac{29x}{5} me \frac{29x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-40.6y=0$$

Tāpiri y ki te -\frac{203y}{5}.

$$-39.6y=0$$

Whakawehea ngā taha e rua o te whārite ki te -39.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki -x+7y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x=0$$

Whakawehea ngā taha e rua ki te -1.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$