Whakaoti i te {l}{x+y=10}{x*5=y}

$$\left. \begin{array} { l } { x + y = 10 } \\ { x \times 5 = y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y = \frac{25}{3} = 8\frac{1}{3} \approx 8.333333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x\times 5-y=0$$

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.

$$x+y=10,5x-y=0$$

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

$$x+y=10$$

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

$$x=-y+10$$

Whakakapia te -y+10 mō te x ki tērā atu whārite, 5x-y=0.

$$5\left(-y+10\right)-y=0$$

Whakareatia 5 ki te -y+10.

$$-5y+50-y=0$$

Tāpiri -5y ki te -y.

$$-6y+50=0$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$-6y=-50$$

Whakawehea ngā taha e rua ki te -6.

$$y=\frac{25}{3}$$

Whakaurua te \frac{25}{3} mō y ki x=-y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{25}{3}+10$$

Tāpiri 10 ki te -\frac{25}{3}.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{25}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x\times 5-y=0$$

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

$$x+y=10,5x-y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\5&-1\end{matrix}\right))\left(\begin{matrix}1&1\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&-1\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&-1\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&-1\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-5}&-\frac{1}{-1-5}\\-\frac{5}{-1-5}&\frac{1}{-1-5}\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{5}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 10\\\frac{5}{6}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\\\frac{25}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{3},y=\frac{25}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x\times 5-y=0$$

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.

$$x+y=10,5x-y=0$$

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

$$5x+5y=5\times 10,5x-y=0$$

Whakarūnātia.

$$5x+5y=50,5x-y=0$$

Me tango 5x-y=0 mai i 5x+5y=50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+5y+y=50$$

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.

$$5y+y=50$$

Tāpiri 5y ki te y.

$$6y=50$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{25}{3}$$

Whakaurua te \frac{25}{3} mō y ki 5x-y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\frac{25}{3}=0$$

Me tāpiri \frac{25}{3} ki ngā taha e rua o te whārite.

$$5x=\frac{25}{3}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{25}{3}$$