Whakaoti i te {l}{y=-5x+6}{y=3x-2}

$$\left\{ \begin{array} { l } { y = - 5 x + 6 } \\ { y = 3 x - 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=6$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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.

$$y+5x=6,y-3x=-2$$

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

$$y+5x=6$$

Me tango 5x mai i ngā taha e rua o te whārite.

$$y=-5x+6$$

Whakakapia te -5x+6 mō te y ki tērā atu whārite, y-3x=-2.

$$-5x+6-3x=-2$$

Tāpiri -5x ki te -3x.

$$-8x+6=-2$$

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

$$-8x=-8$$

Whakawehea ngā taha e rua ki te -8.

$$x=1$$

Whakaurua te 1 mō x ki y=-5x+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-5+6$$

Tāpiri 6 ki te -5.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=6$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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

$$y+5x=6,y-3x=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&5\\1&-3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&5\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&5\\1&-3\end{matrix}\right))\left(\begin{matrix}1&5\\1&-3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-3\end{matrix}\right))\left(\begin{matrix}6\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-3\end{matrix}\right))\left(\begin{matrix}6\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\1&-3\end{matrix}\right))\left(\begin{matrix}6\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-5}&-\frac{5}{-3-5}\\-\frac{1}{-3-5}&\frac{1}{-3-5}\end{matrix}\right)\left(\begin{matrix}6\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&\frac{5}{8}\\\frac{1}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}6\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 6+\frac{5}{8}\left(-2\right)\\\frac{1}{8}\times 6-\frac{1}{8}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=1,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=6$$

Whakaarohia te whārite tuarua. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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.

$$y+5x=6,y-3x=-2$$

Me tango y-3x=-2 mai i y+5x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+5x+3x=6+2$$

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.

$$5x+3x=6+2$$

Tāpiri 5x ki te 3x.

$$8x=6+2$$

Tāpiri 6 ki te 2.

$$8x=8$$

Whakawehea ngā taha e rua ki te 8.

$$x=1$$

Whakaurua te 1 mō x ki y-3x=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-3=-2$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$