Whakaoti i te {l}{y=4x+5}{y=8x+9}

$$\left\{ \begin{array} { l } { y = 4 x + 5 } \\ { y = 8 x + 9 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

Whakaarohia te whārite tuarua. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=9$$

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.

$$y-4x=5,y-8x=9$$

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

$$y-4x=5$$

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

$$y=4x+5$$

Whakakapia te 4x+5 mō te y ki tērā atu whārite, y-8x=9.

$$4x+5-8x=9$$

Tāpiri 4x ki te -8x.

$$-4x+5=9$$

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

$$-4x=4$$

Whakawehea ngā taha e rua ki te -4.

$$x=-1$$

Whakaurua te -1 mō x ki y=4x+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

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

Whakareatia 4 ki te -1.

$$y=-4+5$$

Tāpiri 5 ki te -4.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

Whakaarohia te whārite tuarua. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=9$$

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

$$y-4x=5,y-8x=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right))\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\1&-8\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-\left(-4\right)}&-\frac{-4}{-8-\left(-4\right)}\\-\frac{1}{-8-\left(-4\right)}&\frac{1}{-8-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}5\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2&-1\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}5\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\times 5-9\\\frac{1}{4}\times 5-\frac{1}{4}\times 9\end{matrix}\right)$$

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. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

Whakaarohia te whārite tuarua. Tangohia te 8x mai i ngā taha e rua.

$$y-8x=9$$

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.

$$y-4x=5,y-8x=9$$

Me tango y-8x=9 mai i y-4x=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-4x+8x=5-9$$

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.

$$-4x+8x=5-9$$

Tāpiri -4x ki te 8x.

$$4x=5-9$$

Tāpiri 5 ki te -9.

$$4x=-4$$

Whakawehea ngā taha e rua ki te 4.

$$x=-1$$

Whakaurua te -1 mō x ki y-8x=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-8\left(-1\right)=9$$

Whakareatia -8 ki te -1.

$$y+8=9$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-1$$