Whakaoti i te {l}{y=9x+6}{y=x+7}

$$\left. \begin{array} { l } { y = 9 x + 6 } \\ { y = x + 7 } \end{array} \right.$$

Whakaoti mō y, x

$$x=\frac{1}{8}=0.125$$

$$y = \frac{57}{8} = 7\frac{1}{8} = 7.125$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$y-9x=6$$

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

$$y-x=7$$

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.

$$y-9x=6,y-x=7$$

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

$$y-9x=6$$

Me tāpiri 9x ki ngā taha e rua o te whārite.

$$y=9x+6$$

Whakakapia te 9x+6 mō te y ki tērā atu whārite, y-x=7.

$$9x+6-x=7$$

Tāpiri 9x ki te -x.

$$8x+6=7$$

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

$$8x=1$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}$$

Whakaurua te \frac{1}{8} mō x ki y=9x+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=9\times \frac{1}{8}+6$$

Whakareatia 9 ki te \frac{1}{8}.

$$y=\frac{9}{8}+6$$

Tāpiri 6 ki te \frac{9}{8}.

$$y=\frac{57}{8}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{57}{8},x=\frac{1}{8}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$y-9x=6$$

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

$$y-x=7$$

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

$$y-9x=6,y-x=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-9\\1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right))\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-9\\1&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-9\right)}&-\frac{-9}{-1-\left(-9\right)}\\-\frac{1}{-1-\left(-9\right)}&\frac{1}{-1-\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{8}&\frac{9}{8}\\-\frac{1}{8}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{8}\times 6+\frac{9}{8}\times 7\\-\frac{1}{8}\times 6+\frac{1}{8}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{57}{8}\\\frac{1}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{57}{8},x=\frac{1}{8}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$y-9x=6$$

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

$$y-x=7$$

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.

$$y-9x=6,y-x=7$$

Me tango y-x=7 mai i y-9x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-9x+x=6-7$$

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.

$$-9x+x=6-7$$

Tāpiri -9x ki te x.

$$-8x=6-7$$

Tāpiri 6 ki te -7.

$$-8x=-1$$

Whakawehea ngā taha e rua ki te -8.

$$x=\frac{1}{8}$$

Whakaurua te \frac{1}{8} mō x ki y-x=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{1}{8}=7$$

Me tāpiri \frac{1}{8} ki ngā taha e rua o te whārite.

$$y=\frac{57}{8}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{57}{8},x=\frac{1}{8}$$