Whakaoti i te {l}{text{onmethod.}5x-2y=4}{x=y-2}

$$\left. \begin{array} { l } { \text { on method. } 5 x - 2 y = 4 } \\ { x = y - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$y = \frac{14}{3} = 4\frac{2}{3} \approx 4.666666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-y=-2$$

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.

$$5x-2y=4,x-y=-2$$

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

$$5x-2y=4$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$5x=2y+4$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+4\right)$$

Whakareatia \frac{1}{5} ki te 4+2y.

$$x=\frac{2}{5}y+\frac{4}{5}$$

Whakakapia te \frac{4+2y}{5} mō te x ki tērā atu whārite, x-y=-2.

$$\frac{2}{5}y+\frac{4}{5}-y=-2$$

Tāpiri \frac{2y}{5} ki te -y.

$$-\frac{3}{5}y+\frac{4}{5}=-2$$

Me tango \frac{4}{5} mai i ngā taha e rua o te whārite.

$$-\frac{3}{5}y=-\frac{14}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{14}{3}$$

Whakaurua te \frac{14}{3} mō y ki x=\frac{2}{5}y+\frac{4}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times \frac{14}{3}+\frac{4}{5}$$

Whakareatia \frac{2}{5} ki te \frac{14}{3} 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.

$$x=\frac{28}{15}+\frac{4}{5}$$

Tāpiri \frac{4}{5} ki te \frac{28}{15} 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.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=\frac{14}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-y=-2$$

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

$$5x-2y=4,x-y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\1&-1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5\left(-1\right)-\left(-2\right)}&-\frac{-2}{5\left(-1\right)-\left(-2\right)}\\-\frac{1}{5\left(-1\right)-\left(-2\right)}&\frac{5}{5\left(-1\right)-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}4\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{2}{3}\\\frac{1}{3}&-\frac{5}{3}\end{matrix}\right)\left(\begin{matrix}4\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 4-\frac{2}{3}\left(-2\right)\\\frac{1}{3}\times 4-\frac{5}{3}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\\\frac{14}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{3},y=\frac{14}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-y=-2$$

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.

$$5x-2y=4,x-y=-2$$

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

$$5x-2y=4,5x+5\left(-1\right)y=5\left(-2\right)$$

Whakarūnātia.

$$5x-2y=4,5x-5y=-10$$

Me tango 5x-5y=-10 mai i 5x-2y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-2y+5y=4+10$$

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.

$$-2y+5y=4+10$$

Tāpiri -2y ki te 5y.

$$3y=4+10$$

Tāpiri 4 ki te 10.

$$3y=14$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{14}{3}$$

Whakaurua te \frac{14}{3} mō y ki x-y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{14}{3}=-2$$

Me tāpiri \frac{14}{3} ki ngā taha e rua o te whārite.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=\frac{14}{3}$$