Whakaoti i te {c}{0.4(3x+1)-0.2(2x+y)=-0.4}{3(0.4x-0.5)+5(0.3y-1.1)=-2.8}

$$\left\{ \begin{array} { c } { 0.4 ( 3 x + 1 ) - 0.2 ( 2 x + y ) = - 0.4 } \\ { 3 ( 0.4 x - 0.5 ) + 5 ( 0.3 y - 1.1 ) = - 2.8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-0.25$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 0.4 ki te 3x+1.

$$1.2x+0.4-0.2\left(2x+y\right)=-0.4$$

Whakamahia te āhuatanga tohatoha hei whakarea te -0.2 ki te 2x+y.

$$1.2x+0.4-0.4x-0.2y=-0.4$$

Pahekotia te 1.2x me -0.4x, ka 0.8x.

$$0.8x+0.4-0.2y=-0.4$$

Tangohia te 0.4 mai i ngā taha e rua.

$$0.8x-0.2y=-0.4-0.4$$

Tangohia te 0.4 i te -0.4, ka -0.8.

$$0.8x-0.2y=-0.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 0.4x-0.5.

$$1.2x-1.5+5\left(0.3y-1.1\right)=-2.8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 0.3y-1.1.

$$1.2x-1.5+1.5y-5.5=-2.8$$

Tangohia te 5.5 i te -1.5, ka -7.

$$1.2x-7+1.5y=-2.8$$

Me tāpiri te 7 ki ngā taha e rua.

$$1.2x+1.5y=-2.8+7$$

Tāpirihia te -2.8 ki te 7, ka 4.2.

$$1.2x+1.5y=4.2$$

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.

$$0.8x-0.2y=-0.8,1.2x+1.5y=4.2$$

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

$$0.8x-0.2y=-0.8$$

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

$$0.8x=0.2y-0.8$$

Whakawehea ngā taha e rua o te whārite ki te 0.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=1.25\left(0.2y-0.8\right)$$

Whakareatia 1.25 ki te \frac{y-4}{5}.

$$x=0.25y-1$$

Whakakapia te \frac{y}{4}-1 mō te x ki tērā atu whārite, 1.2x+1.5y=4.2.

$$1.2\left(0.25y-1\right)+1.5y=4.2$$

Whakareatia 1.2 ki te \frac{y}{4}-1.

$$0.3y-1.2+1.5y=4.2$$

Tāpiri \frac{3y}{10} ki te \frac{3y}{2}.

$$1.8y-1.2=4.2$$

Me tāpiri 1.2 ki ngā taha e rua o te whārite.

$$1.8y=5.4$$

Whakawehea ngā taha e rua o te whārite ki te 1.8, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=0.25y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0.25\times 3-1$$

Whakareatia 0.25 ki te 3.

$$x=0.75-1$$

Tāpiri -1 ki te 0.75.

$$x=-0.25$$

Kua oti te pūnaha te whakatau.

$$x=-0.25,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 0.4 ki te 3x+1.

$$1.2x+0.4-0.2\left(2x+y\right)=-0.4$$

Whakamahia te āhuatanga tohatoha hei whakarea te -0.2 ki te 2x+y.

$$1.2x+0.4-0.4x-0.2y=-0.4$$

Pahekotia te 1.2x me -0.4x, ka 0.8x.

$$0.8x+0.4-0.2y=-0.4$$

Tangohia te 0.4 mai i ngā taha e rua.

$$0.8x-0.2y=-0.4-0.4$$

Tangohia te 0.4 i te -0.4, ka -0.8.

$$0.8x-0.2y=-0.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 0.4x-0.5.

$$1.2x-1.5+5\left(0.3y-1.1\right)=-2.8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 0.3y-1.1.

$$1.2x-1.5+1.5y-5.5=-2.8$$

Tangohia te 5.5 i te -1.5, ka -7.

$$1.2x-7+1.5y=-2.8$$

Me tāpiri te 7 ki ngā taha e rua.

$$1.2x+1.5y=-2.8+7$$

Tāpirihia te -2.8 ki te 7, ka 4.2.

$$1.2x+1.5y=4.2$$

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

$$0.8x-0.2y=-0.8,1.2x+1.5y=4.2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right))\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right))\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right))\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.8&-0.2\\1.2&1.5\end{matrix}\right))\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1.5}{0.8\times 1.5-\left(-0.2\times 1.2\right)}&-\frac{-0.2}{0.8\times 1.5-\left(-0.2\times 1.2\right)}\\-\frac{1.2}{0.8\times 1.5-\left(-0.2\times 1.2\right)}&\frac{0.8}{0.8\times 1.5-\left(-0.2\times 1.2\right)}\end{matrix}\right)\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{24}&\frac{5}{36}\\-\frac{5}{6}&\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}-0.8\\4.2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{24}\left(-0.8\right)+\frac{5}{36}\times 4.2\\-\frac{5}{6}\left(-0.8\right)+\frac{5}{9}\times 4.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.25\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-0.25,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 0.4 ki te 3x+1.

$$1.2x+0.4-0.2\left(2x+y\right)=-0.4$$

Whakamahia te āhuatanga tohatoha hei whakarea te -0.2 ki te 2x+y.

$$1.2x+0.4-0.4x-0.2y=-0.4$$

Pahekotia te 1.2x me -0.4x, ka 0.8x.

$$0.8x+0.4-0.2y=-0.4$$

Tangohia te 0.4 mai i ngā taha e rua.

$$0.8x-0.2y=-0.4-0.4$$

Tangohia te 0.4 i te -0.4, ka -0.8.

$$0.8x-0.2y=-0.8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 0.4x-0.5.

$$1.2x-1.5+5\left(0.3y-1.1\right)=-2.8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 0.3y-1.1.

$$1.2x-1.5+1.5y-5.5=-2.8$$

Tangohia te 5.5 i te -1.5, ka -7.

$$1.2x-7+1.5y=-2.8$$

Me tāpiri te 7 ki ngā taha e rua.

$$1.2x+1.5y=-2.8+7$$

Tāpirihia te -2.8 ki te 7, ka 4.2.

$$1.2x+1.5y=4.2$$

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.

$$0.8x-0.2y=-0.8,1.2x+1.5y=4.2$$

Kia ōrite ai a \frac{4x}{5} me \frac{6x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1.2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.8.

$$1.2\times 0.8x+1.2\left(-0.2\right)y=1.2\left(-0.8\right),0.8\times 1.2x+0.8\times 1.5y=0.8\times 4.2$$

Whakarūnātia.

$$0.96x-0.24y=-0.96,0.96x+1.2y=3.36$$

Me tango 0.96x+1.2y=3.36 mai i 0.96x-0.24y=-0.96 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.96x-0.96x-0.24y-1.2y=\frac{-24-84}{25}$$

Tāpiri \frac{24x}{25} ki te -\frac{24x}{25}. Ka whakakore atu ngā kupu \frac{24x}{25} me -\frac{24x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.24y-1.2y=\frac{-24-84}{25}$$

Tāpiri -\frac{6y}{25} ki te -\frac{6y}{5}.

$$-1.44y=\frac{-24-84}{25}$$

Tāpiri -0.96 ki te -3.36 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-1.44y=-4.32$$

Whakawehea ngā taha e rua o te whārite ki te -1.44, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki 1.2x+1.5y=4.2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$1.2x+1.5\times 3=4.2$$

Whakareatia 1.5 ki te 3.

$$1.2x+4.5=4.2$$

Me tango 4.5 mai i ngā taha e rua o te whārite.

$$1.2x=-0.3$$

Whakawehea ngā taha e rua o te whārite ki te 1.2, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-0.25$$

Kua oti te pūnaha te whakatau.

$$x=-0.25,y=3$$