Whakaoti i te {l}{2x+7y/3+y=0}{x+5y-1/2=2-x}

$$\left\{ \begin{array} { l } { \frac { 2 x + 7 y } { 3 } + y = 0 } \\ { x + \frac { 5 y - 1 } { 2 } = 2 - x } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y=-\frac{1}{3}\approx -0.333333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2x+7y+3y=0$$

Pahekotia te 7y me 3y, ka 10y.

$$2x+10y=0$$

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

$$2x+5y-1=4-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$2x+5y-1+2x=4$$

Pahekotia te 2x me 2x, ka 4x.

$$4x+5y-1=4$$

Me tāpiri te 1 ki ngā taha e rua.

$$4x+5y=4+1$$

Tāpirihia te 4 ki te 1, ka 5.

$$4x+5y=5$$

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.

$$2x+10y=0,4x+5y=5$$

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

$$2x+10y=0$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$2x=-10y$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-10\right)y$$

Whakareatia \frac{1}{2} ki te -10y.

$$x=-5y$$

Whakakapia te -5y mō te x ki tērā atu whārite, 4x+5y=5.

$$4\left(-5\right)y+5y=5$$

Whakareatia 4 ki te -5y.

$$-20y+5y=5$$

Tāpiri -20y ki te 5y.

$$-15y=5$$

Whakawehea ngā taha e rua ki te -15.

$$y=-\frac{1}{3}$$

Whakaurua te -\frac{1}{3} mō y ki x=-5y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-\frac{1}{3}\right)$$

Whakareatia -5 ki te -\frac{1}{3}.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=-\frac{1}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2x+7y+3y=0$$

Pahekotia te 7y me 3y, ka 10y.

$$2x+10y=0$$

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

$$2x+5y-1=4-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$2x+5y-1+2x=4$$

Pahekotia te 2x me 2x, ka 4x.

$$4x+5y-1=4$$

Me tāpiri te 1 ki ngā taha e rua.

$$4x+5y=4+1$$

Tāpirihia te 4 ki te 1, ka 5.

$$4x+5y=5$$

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

$$2x+10y=0,4x+5y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&10\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&10\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&10\\4&5\end{matrix}\right))\left(\begin{matrix}2&10\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&10\\4&5\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&10\\4&5\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&10\\4&5\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-10\times 4}&-\frac{10}{2\times 5-10\times 4}\\-\frac{4}{2\times 5-10\times 4}&\frac{2}{2\times 5-10\times 4}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{3}\\\frac{2}{15}&-\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 5\\-\frac{1}{15}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\\-\frac{1}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{3},y=-\frac{1}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2x+7y+3y=0$$

Pahekotia te 7y me 3y, ka 10y.

$$2x+10y=0$$

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

$$2x+5y-1=4-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$2x+5y-1+2x=4$$

Pahekotia te 2x me 2x, ka 4x.

$$4x+5y-1=4$$

Me tāpiri te 1 ki ngā taha e rua.

$$4x+5y=4+1$$

Tāpirihia te 4 ki te 1, ka 5.

$$4x+5y=5$$

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.

$$2x+10y=0,4x+5y=5$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\times 10y=0,2\times 4x+2\times 5y=2\times 5$$

Whakarūnātia.

$$8x+40y=0,8x+10y=10$$

Me tango 8x+10y=10 mai i 8x+40y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+40y-10y=-10$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40y-10y=-10$$

Tāpiri 40y ki te -10y.

$$30y=-10$$

Whakawehea ngā taha e rua ki te 30.

$$y=-\frac{1}{3}$$

Whakaurua te -\frac{1}{3} mō y ki 4x+5y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\left(-\frac{1}{3}\right)=5$$

Whakareatia 5 ki te -\frac{1}{3}.

$$4x-\frac{5}{3}=5$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$4x=\frac{20}{3}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=-\frac{1}{3}$$