Whakaoti i te {l}{5x-y=5}{y=1/5x}

$$\left\{ \begin{array} { l } { 5 x - y = 5 } \\ { y = \frac { 1 } { 5 } x } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{25}{24} = 1\frac{1}{24} \approx 1.041666667$$

$$y=\frac{5}{24}\approx 0.208333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{5}x mai i ngā taha e rua.

$$y-\frac{1}{5}x=0$$

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.

$$5x-y=5,-\frac{1}{5}x+y=0$$

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

$$5x-y=5$$

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

$$5x=y+5$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+5\right)$$

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

$$x=\frac{1}{5}y+1$$

Whakakapia te \frac{y}{5}+1 mō te x ki tērā atu whārite, -\frac{1}{5}x+y=0.

$$-\frac{1}{5}\left(\frac{1}{5}y+1\right)+y=0$$

Whakareatia -\frac{1}{5} ki te \frac{y}{5}+1.

$$-\frac{1}{25}y-\frac{1}{5}+y=0$$

Tāpiri -\frac{y}{25} ki te y.

$$\frac{24}{25}y-\frac{1}{5}=0$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$\frac{24}{25}y=\frac{1}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{24}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5}{24}$$

Whakaurua te \frac{5}{24} mō y ki x=\frac{1}{5}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times \frac{5}{24}+1$$

Whakareatia \frac{1}{5} ki te \frac{5}{24} 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.

$$x=\frac{1}{24}+1$$

Tāpiri 1 ki te \frac{1}{24}.

$$x=\frac{25}{24}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{24},y=\frac{5}{24}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{5}x mai i ngā taha e rua.

$$y-\frac{1}{5}x=0$$

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

$$5x-y=5,-\frac{1}{5}x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\-\frac{1}{5}&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{24}&\frac{5}{24}\\\frac{1}{24}&\frac{25}{24}\end{matrix}\right)\left(\begin{matrix}5\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{24}\times 5\\\frac{1}{24}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{24}\\\frac{5}{24}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{25}{24},y=\frac{5}{24}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{5}x mai i ngā taha e rua.

$$y-\frac{1}{5}x=0$$

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.

$$5x-y=5,-\frac{1}{5}x+y=0$$

Kia ōrite ai a 5x me -\frac{x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -\frac{1}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-\frac{1}{5}\times 5x-\frac{1}{5}\left(-1\right)y=-\frac{1}{5}\times 5,5\left(-\frac{1}{5}\right)x+5y=0$$

Whakarūnātia.

$$-x+\frac{1}{5}y=-1,-x+5y=0$$

Me tango -x+5y=0 mai i -x+\frac{1}{5}y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x+\frac{1}{5}y-5y=-1$$

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

$$\frac{1}{5}y-5y=-1$$

Tāpiri \frac{y}{5} ki te -5y.

$$-\frac{24}{5}y=-1$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{24}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5}{24}$$

Whakaurua te \frac{5}{24} mō y ki -\frac{1}{5}x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-\frac{1}{5}x+\frac{5}{24}=0$$

Me tango \frac{5}{24} mai i ngā taha e rua o te whārite.

$$-\frac{1}{5}x=-\frac{5}{24}$$

Me whakarea ngā taha e rua ki te -5.

$$x=\frac{25}{24}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{24},y=\frac{5}{24}$$