Whakaoti i te {l}{y=2/-7x+3468}{y=7/2x-248}

$$\left. \begin{array} { l } { y = \frac { 2 } { - 7 } x + 3468 } \\ { y = \frac { 7 } { 2 } x - 248 } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{52024}{53} = 981\frac{31}{53} \approx 981.58490566$$

$$y = \frac{168940}{53} = 3187\frac{29}{53} \approx 3187.547169811$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{2}{-7} te tuhi anō ko -\frac{2}{7} mā te tango i te tohu tōraro.

$$y=-\frac{2}{7}x+3468$$

Whakakapia te -\frac{2x}{7}+3468 mō te y ki tērā atu whārite, y-\frac{7}{2}x=-248.

$$-\frac{2}{7}x+3468-\frac{7}{2}x=-248$$

Tāpiri -\frac{2x}{7} ki te -\frac{7x}{2}.

$$-\frac{53}{14}x+3468=-248$$

Me tango 3468 mai i ngā taha e rua o te whārite.

$$-\frac{53}{14}x=-3716$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{53}{14}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{52024}{53}$$

Whakaurua te \frac{52024}{53} mō x ki y=-\frac{2}{7}x+3468. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{2}{7}\times \frac{52024}{53}+3468$$

Whakareatia -\frac{2}{7} ki te \frac{52024}{53} 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.

$$y=-\frac{14864}{53}+3468$$

Tāpiri 3468 ki te -\frac{14864}{53}.

$$y=\frac{168940}{53}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{168940}{53},x=\frac{52024}{53}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{2}{-7} te tuhi anō ko -\frac{2}{7} mā te tango i te tohu tōraro.

$$y=-\frac{2}{7}x+3468$$

Me tāpiri te \frac{2}{7}x ki ngā taha e rua.

$$y+\frac{2}{7}x=3468$$

Whakaarohia te whārite tuarua. Tangohia te \frac{7}{2}x mai i ngā taha e rua.

$$y-\frac{7}{2}x=-248$$

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

$$y+\frac{2}{7}x=3468,y-\frac{7}{2}x=-248$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right))\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right))\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right))\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{2}{7}\\1&-\frac{7}{2}\end{matrix}\right))\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{7}{2}}{-\frac{7}{2}-\frac{2}{7}}&-\frac{\frac{2}{7}}{-\frac{7}{2}-\frac{2}{7}}\\-\frac{1}{-\frac{7}{2}-\frac{2}{7}}&\frac{1}{-\frac{7}{2}-\frac{2}{7}}\end{matrix}\right)\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{49}{53}&\frac{4}{53}\\\frac{14}{53}&-\frac{14}{53}\end{matrix}\right)\left(\begin{matrix}3468\\-248\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{49}{53}\times 3468+\frac{4}{53}\left(-248\right)\\\frac{14}{53}\times 3468-\frac{14}{53}\left(-248\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{168940}{53}\\\frac{52024}{53}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{168940}{53},x=\frac{52024}{53}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Ka taea te hautanga \frac{2}{-7} te tuhi anō ko -\frac{2}{7} mā te tango i te tohu tōraro.

$$y=-\frac{2}{7}x+3468$$

Me tāpiri te \frac{2}{7}x ki ngā taha e rua.

$$y+\frac{2}{7}x=3468$$

Whakaarohia te whārite tuarua. Tangohia te \frac{7}{2}x mai i ngā taha e rua.

$$y-\frac{7}{2}x=-248$$

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.

$$y+\frac{2}{7}x=3468,y-\frac{7}{2}x=-248$$

Me tango y-\frac{7}{2}x=-248 mai i y+\frac{2}{7}x=3468 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\frac{2}{7}x+\frac{7}{2}x=3468+248$$

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.

$$\frac{2}{7}x+\frac{7}{2}x=3468+248$$

Tāpiri \frac{2x}{7} ki te \frac{7x}{2}.

$$\frac{53}{14}x=3468+248$$

Tāpiri 3468 ki te 248.

$$\frac{53}{14}x=3716$$

Whakawehea ngā taha e rua o te whārite ki te \frac{53}{14}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{52024}{53}$$

Whakaurua te \frac{52024}{53} mō x ki y-\frac{7}{2}x=-248. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{7}{2}\times \frac{52024}{53}=-248$$

Whakareatia -\frac{7}{2} ki te \frac{52024}{53} 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.

$$y-\frac{182084}{53}=-248$$

Me tāpiri \frac{182084}{53} ki ngā taha e rua o te whārite.

$$y=\frac{168940}{53}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{168940}{53},x=\frac{52024}{53}$$