Whakaoti i te {l}{x+y=204}{2/3y=3/4x}

$$\left\{ \begin{array} { l } { x + y = 204 } \\ { \frac { 2 } { 3 } y = \frac { 3 } { 4 } x } \end{array} \right.$$

Whakaoti mō x, y

$$x=96$$

$$y=108$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{4}x mai i ngā taha e rua.

$$\frac{2}{3}y-\frac{3}{4}x=0$$

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.

$$x+y=204,-\frac{3}{4}x+\frac{2}{3}y=0$$

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

$$x+y=204$$

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

$$x=-y+204$$

Whakakapia te -y+204 mō te x ki tērā atu whārite, -\frac{3}{4}x+\frac{2}{3}y=0.

$$-\frac{3}{4}\left(-y+204\right)+\frac{2}{3}y=0$$

Whakareatia -\frac{3}{4} ki te -y+204.

$$\frac{3}{4}y-153+\frac{2}{3}y=0$$

Tāpiri \frac{3y}{4} ki te \frac{2y}{3}.

$$\frac{17}{12}y-153=0$$

Me tāpiri 153 ki ngā taha e rua o te whārite.

$$\frac{17}{12}y=153$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{12}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=108$$

Whakaurua te 108 mō y ki x=-y+204. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-108+204$$

Tāpiri 204 ki te -108.

$$x=96$$

Kua oti te pūnaha te whakatau.

$$x=96,y=108$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{4}x mai i ngā taha e rua.

$$\frac{2}{3}y-\frac{3}{4}x=0$$

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

$$x+y=204,-\frac{3}{4}x+\frac{2}{3}y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}204\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}204\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}204\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-\frac{3}{4}&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}204\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{2}{3}}{\frac{2}{3}-\left(-\frac{3}{4}\right)}&-\frac{1}{\frac{2}{3}-\left(-\frac{3}{4}\right)}\\-\frac{-\frac{3}{4}}{\frac{2}{3}-\left(-\frac{3}{4}\right)}&\frac{1}{\frac{2}{3}-\left(-\frac{3}{4}\right)}\end{matrix}\right)\left(\begin{matrix}204\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{17}&-\frac{12}{17}\\\frac{9}{17}&\frac{12}{17}\end{matrix}\right)\left(\begin{matrix}204\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{17}\times 204\\\frac{9}{17}\times 204\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}96\\108\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=96,y=108$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{4}x mai i ngā taha e rua.

$$\frac{2}{3}y-\frac{3}{4}x=0$$

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.

$$x+y=204,-\frac{3}{4}x+\frac{2}{3}y=0$$

Kia ōrite ai a x me -\frac{3x}{4}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -\frac{3}{4} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-\frac{3}{4}x-\frac{3}{4}y=-\frac{3}{4}\times 204,-\frac{3}{4}x+\frac{2}{3}y=0$$

Whakarūnātia.

$$-\frac{3}{4}x-\frac{3}{4}y=-153,-\frac{3}{4}x+\frac{2}{3}y=0$$

Me tango -\frac{3}{4}x+\frac{2}{3}y=0 mai i -\frac{3}{4}x-\frac{3}{4}y=-153 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-\frac{3}{4}x+\frac{3}{4}x-\frac{3}{4}y-\frac{2}{3}y=-153$$

Tāpiri -\frac{3x}{4} ki te \frac{3x}{4}. Ka whakakore atu ngā kupu -\frac{3x}{4} me \frac{3x}{4}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-\frac{3}{4}y-\frac{2}{3}y=-153$$

Tāpiri -\frac{3y}{4} ki te -\frac{2y}{3}.

$$-\frac{17}{12}y=-153$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{12}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=108$$

Whakaurua te 108 mō y ki -\frac{3}{4}x+\frac{2}{3}y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-\frac{3}{4}x+\frac{2}{3}\times 108=0$$

Whakareatia \frac{2}{3} ki te 108.

$$-\frac{3}{4}x+72=0$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-\frac{3}{4}x=-72$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=96$$

Kua oti te pūnaha te whakatau.

$$x=96,y=108$$