Whakaoti i te {l}{3(x+y)-4(x-y)=-18}{1/2(x+y)+1/6(x-y)=2}

$$\left\{ \begin{array} { l } { 3 ( x + y ) - 4 ( x - y ) = - 18 } \\ { \frac { 1 } { 2 } ( x + y ) + \frac { 1 } { 6 } ( x - y ) = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

$$3x+3y-4\left(x-y\right)=-18$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te x-y.

$$3x+3y-4x+4y=-18$$

Pahekotia te 3x me -4x, ka -x.

$$-x+3y+4y=-18$$

Pahekotia te 3y me 4y, ka 7y.

$$-x+7y=-18$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2} ki te x+y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}\left(x-y\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{6} ki te x-y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}x-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}x me \frac{1}{6}x, ka \frac{2}{3}x.

$$\frac{2}{3}x+\frac{1}{2}y-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}y me -\frac{1}{6}y, ka \frac{1}{3}y.

$$\frac{2}{3}x+\frac{1}{3}y=2$$

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.

$$-x+7y=-18,\frac{2}{3}x+\frac{1}{3}y=2$$

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

$$-x+7y=-18$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$-x=-7y-18$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-7y-18\right)$$

Whakareatia -1 ki te -7y-18.

$$x=7y+18$$

Whakakapia te 7y+18 mō te x ki tērā atu whārite, \frac{2}{3}x+\frac{1}{3}y=2.

$$\frac{2}{3}\left(7y+18\right)+\frac{1}{3}y=2$$

Whakareatia \frac{2}{3} ki te 7y+18.

$$\frac{14}{3}y+12+\frac{1}{3}y=2$$

Tāpiri \frac{14y}{3} ki te \frac{y}{3}.

$$5y+12=2$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$5y=-10$$

Whakawehea ngā taha e rua ki te 5.

$$y=-2$$

Whakaurua te -2 mō y ki x=7y+18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=7\left(-2\right)+18$$

Whakareatia 7 ki te -2.

$$x=-14+18$$

Tāpiri 18 ki te -14.

$$x=4$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

$$3x+3y-4\left(x-y\right)=-18$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te x-y.

$$3x+3y-4x+4y=-18$$

Pahekotia te 3x me -4x, ka -x.

$$-x+3y+4y=-18$$

Pahekotia te 3y me 4y, ka 7y.

$$-x+7y=-18$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2} ki te x+y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}\left(x-y\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{6} ki te x-y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}x-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}x me \frac{1}{6}x, ka \frac{2}{3}x.

$$\frac{2}{3}x+\frac{1}{2}y-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}y me -\frac{1}{6}y, ka \frac{1}{3}y.

$$\frac{2}{3}x+\frac{1}{3}y=2$$

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

$$-x+7y=-18,\frac{2}{3}x+\frac{1}{3}y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-18\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{3}}{-\frac{1}{3}-7\times \frac{2}{3}}&-\frac{7}{-\frac{1}{3}-7\times \frac{2}{3}}\\-\frac{\frac{2}{3}}{-\frac{1}{3}-7\times \frac{2}{3}}&-\frac{1}{-\frac{1}{3}-7\times \frac{2}{3}}\end{matrix}\right)\left(\begin{matrix}-18\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}&\frac{7}{5}\\\frac{2}{15}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-18\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{15}\left(-18\right)+\frac{7}{5}\times 2\\\frac{2}{15}\left(-18\right)+\frac{1}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+y.

$$3x+3y-4\left(x-y\right)=-18$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te x-y.

$$3x+3y-4x+4y=-18$$

Pahekotia te 3x me -4x, ka -x.

$$-x+3y+4y=-18$$

Pahekotia te 3y me 4y, ka 7y.

$$-x+7y=-18$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2} ki te x+y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}\left(x-y\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{6} ki te x-y.

$$\frac{1}{2}x+\frac{1}{2}y+\frac{1}{6}x-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}x me \frac{1}{6}x, ka \frac{2}{3}x.

$$\frac{2}{3}x+\frac{1}{2}y-\frac{1}{6}y=2$$

Pahekotia te \frac{1}{2}y me -\frac{1}{6}y, ka \frac{1}{3}y.

$$\frac{2}{3}x+\frac{1}{3}y=2$$

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.

$$-x+7y=-18,\frac{2}{3}x+\frac{1}{3}y=2$$

Kia ōrite ai a -x me \frac{2x}{3}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{2}{3} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$\frac{2}{3}\left(-1\right)x+\frac{2}{3}\times 7y=\frac{2}{3}\left(-18\right),-\frac{2}{3}x-\frac{1}{3}y=-2$$

Whakarūnātia.

$$-\frac{2}{3}x+\frac{14}{3}y=-12,-\frac{2}{3}x-\frac{1}{3}y=-2$$

Me tango -\frac{2}{3}x-\frac{1}{3}y=-2 mai i -\frac{2}{3}x+\frac{14}{3}y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-\frac{2}{3}x+\frac{2}{3}x+\frac{14}{3}y+\frac{1}{3}y=-12+2$$

Tāpiri -\frac{2x}{3} ki te \frac{2x}{3}. Ka whakakore atu ngā kupu -\frac{2x}{3} me \frac{2x}{3}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{14}{3}y+\frac{1}{3}y=-12+2$$

Tāpiri \frac{14y}{3} ki te \frac{y}{3}.

$$5y=-12+2$$

Tāpiri -12 ki te 2.

$$5y=-10$$

Whakawehea ngā taha e rua ki te 5.

$$y=-2$$

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

$$\frac{2}{3}x+\frac{1}{3}\left(-2\right)=2$$

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

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

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$\frac{2}{3}x=\frac{8}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

Kua oti te pūnaha te whakatau.

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