Whakaoti i te {l}{x+y=5y}{x+8y=9}

$$\left. \begin{array} { l } { x + y = 5 y } \\ { x + 8 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=\frac{3}{4}=0.75$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x+y-5y=0$$

Pahekotia te y me -5y, ka -4y.

$$x-4y=0$$

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.

$$x-4y=0,x+8y=9$$

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

$$x-4y=0$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$x=4y$$

Whakakapia te 4y mō te x ki tērā atu whārite, x+8y=9.

$$4y+8y=9$$

Tāpiri 4y ki te 8y.

$$12y=9$$

Whakawehea ngā taha e rua ki te 12.

$$y=\frac{3}{4}$$

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

$$x=4\times \frac{3}{4}$$

Whakareatia 4 ki te \frac{3}{4}.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{3}{4}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x+y-5y=0$$

Pahekotia te y me -5y, ka -4y.

$$x-4y=0$$

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

$$x-4y=0,x+8y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\1&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-4\\1&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-4\\1&8\end{matrix}\right))\left(\begin{matrix}1&-4\\1&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&8\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&8\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&8\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{8-\left(-4\right)}&-\frac{-4}{8-\left(-4\right)}\\-\frac{1}{8-\left(-4\right)}&\frac{1}{8-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}0\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\-\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}0\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 9\\\frac{1}{12}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\\frac{3}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=\frac{3}{4}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x+y-5y=0$$

Pahekotia te y me -5y, ka -4y.

$$x-4y=0$$

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.

$$x-4y=0,x+8y=9$$

Me tango x+8y=9 mai i x-4y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-4y-8y=-9$$

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.

$$-4y-8y=-9$$

Tāpiri -4y ki te -8y.

$$-12y=-9$$

Whakawehea ngā taha e rua ki te -12.

$$y=\frac{3}{4}$$

Whakaurua te \frac{3}{4} mō y ki x+8y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+8\times \frac{3}{4}=9$$

Whakareatia 8 ki te \frac{3}{4}.

$$x+6=9$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{3}{4}$$