Whakaoti i te {l}{5x-4y=19y}{5x+2y=71}

$$\left\{ \begin{array} { l } { 5 x - 4 y = 19 y } \\ { 5 x + 2 y = 71 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{1633}{125} = 13\frac{8}{125} = 13.064$$

$$y = \frac{71}{25} = 2\frac{21}{25} = 2.84$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 19y mai i ngā taha e rua.

$$5x-4y-19y=0$$

Pahekotia te -4y me -19y, ka -23y.

$$5x-23y=0$$

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.

$$5x-23y=0,5x+2y=71$$

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

$$5x-23y=0$$

Me tāpiri 23y ki ngā taha e rua o te whārite.

$$5x=23y$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\times 23y$$

Whakareatia \frac{1}{5} ki te 23y.

$$x=\frac{23}{5}y$$

Whakakapia te \frac{23y}{5} mō te x ki tērā atu whārite, 5x+2y=71.

$$5\times \frac{23}{5}y+2y=71$$

Whakareatia 5 ki te \frac{23y}{5}.

$$23y+2y=71$$

Tāpiri 23y ki te 2y.

$$25y=71$$

Whakawehea ngā taha e rua ki te 25.

$$y=\frac{71}{25}$$

Whakaurua te \frac{71}{25} mō y ki x=\frac{23}{5}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{23}{5}\times \frac{71}{25}$$

Whakareatia \frac{23}{5} ki te \frac{71}{25} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1633}{125}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1633}{125},y=\frac{71}{25}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 19y mai i ngā taha e rua.

$$5x-4y-19y=0$$

Pahekotia te -4y me -19y, ka -23y.

$$5x-23y=0$$

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

$$5x-23y=0,5x+2y=71$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-23\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\71\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-23\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-23\\5&2\end{matrix}\right))\left(\begin{matrix}5&-23\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-23\\5&2\end{matrix}\right))\left(\begin{matrix}0\\71\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-23\\5&2\end{matrix}\right))\left(\begin{matrix}0\\71\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-23\\5&2\end{matrix}\right))\left(\begin{matrix}0\\71\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-23\times 5\right)}&-\frac{-23}{5\times 2-\left(-23\times 5\right)}\\-\frac{5}{5\times 2-\left(-23\times 5\right)}&\frac{5}{5\times 2-\left(-23\times 5\right)}\end{matrix}\right)\left(\begin{matrix}0\\71\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{125}&\frac{23}{125}\\-\frac{1}{25}&\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}0\\71\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{125}\times 71\\\frac{1}{25}\times 71\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1633}{125}\\\frac{71}{25}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1633}{125},y=\frac{71}{25}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 19y mai i ngā taha e rua.

$$5x-4y-19y=0$$

Pahekotia te -4y me -19y, ka -23y.

$$5x-23y=0$$

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.

$$5x-23y=0,5x+2y=71$$

Me tango 5x+2y=71 mai i 5x-23y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-23y-2y=-71$$

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

$$-23y-2y=-71$$

Tāpiri -23y ki te -2y.

$$-25y=-71$$

Whakawehea ngā taha e rua ki te -25.

$$y=\frac{71}{25}$$

Whakaurua te \frac{71}{25} mō y ki 5x+2y=71. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2\times \frac{71}{25}=71$$

Whakareatia 2 ki te \frac{71}{25}.

$$5x+\frac{142}{25}=71$$

Me tango \frac{142}{25} mai i ngā taha e rua o te whārite.

$$5x=\frac{1633}{25}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1633}{125}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1633}{125},y=\frac{71}{25}$$