Whakaoti i te {l}{y=-3/4x+3/4}{y=4/3x+11/3}

$$\left\{ \begin{array} { l } { y = - \frac { 3 } { 4 } x + \frac { 3 } { 4 } } \\ { y = \frac { 4 } { 3 } x + \frac { 11 } { 3 } } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{7}{5} = -1\frac{2}{5} = -1.4$$

$$y = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{3}{4}x ki ngā taha e rua.

$$y+\frac{3}{4}x=\frac{3}{4}$$

Whakaarohia te whārite tuarua. Tangohia te \frac{4}{3}x mai i ngā taha e rua.

$$y-\frac{4}{3}x=\frac{11}{3}$$

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.

$$y+\frac{3}{4}x=\frac{3}{4},y-\frac{4}{3}x=\frac{11}{3}$$

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

$$y+\frac{3}{4}x=\frac{3}{4}$$

Me tango \frac{3x}{4} mai i ngā taha e rua o te whārite.

$$y=-\frac{3}{4}x+\frac{3}{4}$$

Whakakapia te \frac{-3x+3}{4} mō te y ki tērā atu whārite, y-\frac{4}{3}x=\frac{11}{3}.

$$-\frac{3}{4}x+\frac{3}{4}-\frac{4}{3}x=\frac{11}{3}$$

Tāpiri -\frac{3x}{4} ki te -\frac{4x}{3}.

$$-\frac{25}{12}x+\frac{3}{4}=\frac{11}{3}$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$-\frac{25}{12}x=\frac{35}{12}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{25}{12}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{7}{5}$$

Whakaurua te -\frac{7}{5} mō x ki y=-\frac{3}{4}x+\frac{3}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{3}{4}\left(-\frac{7}{5}\right)+\frac{3}{4}$$

Whakareatia -\frac{3}{4} ki te -\frac{7}{5} 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.

$$y=\frac{21}{20}+\frac{3}{4}$$

Tāpiri \frac{3}{4} ki te \frac{21}{20} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{9}{5},x=-\frac{7}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{3}{4}x ki ngā taha e rua.

$$y+\frac{3}{4}x=\frac{3}{4}$$

Whakaarohia te whārite tuarua. Tangohia te \frac{4}{3}x mai i ngā taha e rua.

$$y-\frac{4}{3}x=\frac{11}{3}$$

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

$$y+\frac{3}{4}x=\frac{3}{4},y-\frac{4}{3}x=\frac{11}{3}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right))\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right))\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right))\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{4}\\1&-\frac{4}{3}\end{matrix}\right))\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{4}{3}}{-\frac{4}{3}-\frac{3}{4}}&-\frac{\frac{3}{4}}{-\frac{4}{3}-\frac{3}{4}}\\-\frac{1}{-\frac{4}{3}-\frac{3}{4}}&\frac{1}{-\frac{4}{3}-\frac{3}{4}}\end{matrix}\right)\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{16}{25}&\frac{9}{25}\\\frac{12}{25}&-\frac{12}{25}\end{matrix}\right)\left(\begin{matrix}\frac{3}{4}\\\frac{11}{3}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{16}{25}\times \frac{3}{4}+\frac{9}{25}\times \frac{11}{3}\\\frac{12}{25}\times \frac{3}{4}-\frac{12}{25}\times \frac{11}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{5}\\-\frac{7}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{9}{5},x=-\frac{7}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{3}{4}x ki ngā taha e rua.

$$y+\frac{3}{4}x=\frac{3}{4}$$

Whakaarohia te whārite tuarua. Tangohia te \frac{4}{3}x mai i ngā taha e rua.

$$y-\frac{4}{3}x=\frac{11}{3}$$

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.

$$y+\frac{3}{4}x=\frac{3}{4},y-\frac{4}{3}x=\frac{11}{3}$$

Me tango y-\frac{4}{3}x=\frac{11}{3} mai i y+\frac{3}{4}x=\frac{3}{4} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\frac{3}{4}x+\frac{4}{3}x=\frac{3}{4}-\frac{11}{3}$$

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.

$$\frac{3}{4}x+\frac{4}{3}x=\frac{3}{4}-\frac{11}{3}$$

Tāpiri \frac{3x}{4} ki te \frac{4x}{3}.

$$\frac{25}{12}x=\frac{3}{4}-\frac{11}{3}$$

Tāpiri \frac{3}{4} ki te -\frac{11}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$\frac{25}{12}x=-\frac{35}{12}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{12}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{7}{5}$$

Whakaurua te -\frac{7}{5} mō x ki y-\frac{4}{3}x=\frac{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{4}{3}\left(-\frac{7}{5}\right)=\frac{11}{3}$$

Whakareatia -\frac{4}{3} ki te -\frac{7}{5} 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.

$$y+\frac{28}{15}=\frac{11}{3}$$

Me tango \frac{28}{15} mai i ngā taha e rua o te whārite.

$$y=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{9}{5},x=-\frac{7}{5}$$