Whakaoti i te {l}{x-y=21}{x-y/20=16}

$$\left. \begin{array} { l } { x - y = 21 } \\ { x - \frac { y } { 20 } = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{299}{19} = 15\frac{14}{19} \approx 15.736842105$$

$$y = -\frac{100}{19} = -5\frac{5}{19} \approx -5.263157895$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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.

$$x-y=21,20x-y=320$$

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

$$x-y=21$$

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

$$x=y+21$$

Whakakapia te y+21 mō te x ki tērā atu whārite, 20x-y=320.

$$20\left(y+21\right)-y=320$$

Whakareatia 20 ki te y+21.

$$20y+420-y=320$$

Tāpiri 20y ki te -y.

$$19y+420=320$$

Me tango 420 mai i ngā taha e rua o te whārite.

$$19y=-100$$

Whakawehea ngā taha e rua ki te 19.

$$y=-\frac{100}{19}$$

Whakaurua te -\frac{100}{19} mō y ki x=y+21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{100}{19}+21$$

Tāpiri 21 ki te -\frac{100}{19}.

$$x=\frac{299}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{299}{19},y=-\frac{100}{19}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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

$$x-y=21,20x-y=320$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\320\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\20&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right))\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right))\left(\begin{matrix}21\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right))\left(\begin{matrix}21\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\20&-1\end{matrix}\right))\left(\begin{matrix}21\\320\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-20\right)}&-\frac{-1}{-1-\left(-20\right)}\\-\frac{20}{-1-\left(-20\right)}&\frac{1}{-1-\left(-20\right)}\end{matrix}\right)\left(\begin{matrix}21\\320\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}&\frac{1}{19}\\-\frac{20}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}21\\320\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}\times 21+\frac{1}{19}\times 320\\-\frac{20}{19}\times 21+\frac{1}{19}\times 320\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{299}{19}\\-\frac{100}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{299}{19},y=-\frac{100}{19}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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.

$$x-y=21,20x-y=320$$

Me tango 20x-y=320 mai i x-y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-20x-y+y=21-320$$

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

$$x-20x=21-320$$

Tāpiri x ki te -20x.

$$-19x=21-320$$

Tāpiri 21 ki te -320.

$$-19x=-299$$

Whakawehea ngā taha e rua ki te -19.

$$x=\frac{299}{19}$$

Whakaurua te \frac{299}{19} mō x ki 20x-y=320. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$20\times \frac{299}{19}-y=320$$

Whakareatia 20 ki te \frac{299}{19}.

$$\frac{5980}{19}-y=320$$

Me tango \frac{5980}{19} mai i ngā taha e rua o te whārite.

$$-y=\frac{100}{19}$$

Whakawehea ngā taha e rua ki te -1.

$$y=-\frac{100}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{299}{19},y=-\frac{100}{19}$$