Whakaoti i te {l}{y=1/3x+6}{y=1/9x-1}

$$\left. \begin{array} { l } { y = \frac { 1 } { 3 } x + 6 } \\ { y = \frac { 1 } { 9 } x - 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{63}{2} = -31\frac{1}{2} = -31.5$$

$$y = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

$$y-\frac{1}{3}x=6$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{9}x mai i ngā taha e rua.

$$y-\frac{1}{9}x=-1$$

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.

$$y-\frac{1}{3}x=6,y-\frac{1}{9}x=-1$$

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

$$y-\frac{1}{3}x=6$$

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

$$y=\frac{1}{3}x+6$$

Whakakapia te \frac{x}{3}+6 mō te y ki tērā atu whārite, y-\frac{1}{9}x=-1.

$$\frac{1}{3}x+6-\frac{1}{9}x=-1$$

Tāpiri \frac{x}{3} ki te -\frac{x}{9}.

$$\frac{2}{9}x+6=-1$$

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

$$\frac{2}{9}x=-7$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{63}{2}$$

Whakaurua te -\frac{63}{2} mō x ki y=\frac{1}{3}x+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{3}\left(-\frac{63}{2}\right)+6$$

Whakareatia \frac{1}{3} ki te -\frac{63}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=-\frac{21}{2}+6$$

Tāpiri 6 ki te -\frac{21}{2}.

$$y=-\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{9}{2},x=-\frac{63}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

$$y-\frac{1}{3}x=6$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{9}x mai i ngā taha e rua.

$$y-\frac{1}{9}x=-1$$

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

$$y-\frac{1}{3}x=6,y-\frac{1}{9}x=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right))\left(\begin{matrix}6\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right))\left(\begin{matrix}6\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&-\frac{1}{9}\end{matrix}\right))\left(\begin{matrix}6\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{1}{9}}{-\frac{1}{9}-\left(-\frac{1}{3}\right)}&-\frac{-\frac{1}{3}}{-\frac{1}{9}-\left(-\frac{1}{3}\right)}\\-\frac{1}{-\frac{1}{9}-\left(-\frac{1}{3}\right)}&\frac{1}{-\frac{1}{9}-\left(-\frac{1}{3}\right)}\end{matrix}\right)\left(\begin{matrix}6\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{3}{2}\\-\frac{9}{2}&\frac{9}{2}\end{matrix}\right)\left(\begin{matrix}6\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 6+\frac{3}{2}\left(-1\right)\\-\frac{9}{2}\times 6+\frac{9}{2}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{2}\\-\frac{63}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{9}{2},x=-\frac{63}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te \frac{1}{3}x mai i ngā taha e rua.

$$y-\frac{1}{3}x=6$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{9}x mai i ngā taha e rua.

$$y-\frac{1}{9}x=-1$$

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.

$$y-\frac{1}{3}x=6,y-\frac{1}{9}x=-1$$

Me tango y-\frac{1}{9}x=-1 mai i y-\frac{1}{3}x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-\frac{1}{3}x+\frac{1}{9}x=6+1$$

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.

$$-\frac{1}{3}x+\frac{1}{9}x=6+1$$

Tāpiri -\frac{x}{3} ki te \frac{x}{9}.

$$-\frac{2}{9}x=6+1$$

Tāpiri 6 ki te 1.

$$-\frac{2}{9}x=7$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{63}{2}$$

Whakaurua te -\frac{63}{2} mō x ki y-\frac{1}{9}x=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{1}{9}\left(-\frac{63}{2}\right)=-1$$

Whakareatia -\frac{1}{9} ki te -\frac{63}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y+\frac{7}{2}=-1$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$y=-\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{9}{2},x=-\frac{63}{2}$$