Whakaoti i te {r}{7x+2y=24}{-8x+2y=-30}

$$\left. \begin{array} { r } { 7 x + 2 y = 24 } \\ { - 8 x + 2 y = - 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

$$y=-\frac{3}{5}=-0.6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$7x+2y=24,-8x+2y=-30$$

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

$$7x+2y=24$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$7x=-2y+24$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-2y+24\right)$$

Whakareatia \frac{1}{7} ki te -2y+24.

$$x=-\frac{2}{7}y+\frac{24}{7}$$

Whakakapia te \frac{-2y+24}{7} mō te x ki tērā atu whārite, -8x+2y=-30.

$$-8\left(-\frac{2}{7}y+\frac{24}{7}\right)+2y=-30$$

Whakareatia -8 ki te \frac{-2y+24}{7}.

$$\frac{16}{7}y-\frac{192}{7}+2y=-30$$

Tāpiri \frac{16y}{7} ki te 2y.

$$\frac{30}{7}y-\frac{192}{7}=-30$$

Me tāpiri \frac{192}{7} ki ngā taha e rua o te whārite.

$$\frac{30}{7}y=-\frac{18}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{30}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{3}{5}$$

Whakaurua te -\frac{3}{5} mō y ki x=-\frac{2}{7}y+\frac{24}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{7}\left(-\frac{3}{5}\right)+\frac{24}{7}$$

Whakareatia -\frac{2}{7} ki te -\frac{3}{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.

$$x=\frac{6}{35}+\frac{24}{7}$$

Tāpiri \frac{24}{7} ki te \frac{6}{35} 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.

$$x=\frac{18}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{5},y=-\frac{3}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$7x+2y=24,-8x+2y=-30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&2\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\-30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&2\\-8&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&2\\-8&2\end{matrix}\right))\left(\begin{matrix}7&2\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\-8&2\end{matrix}\right))\left(\begin{matrix}24\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\-8&2\end{matrix}\right))\left(\begin{matrix}24\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\-8&2\end{matrix}\right))\left(\begin{matrix}24\\-30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-2\left(-8\right)}&-\frac{2}{7\times 2-2\left(-8\right)}\\-\frac{-8}{7\times 2-2\left(-8\right)}&\frac{7}{7\times 2-2\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}24\\-30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&-\frac{1}{15}\\\frac{4}{15}&\frac{7}{30}\end{matrix}\right)\left(\begin{matrix}24\\-30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 24-\frac{1}{15}\left(-30\right)\\\frac{4}{15}\times 24+\frac{7}{30}\left(-30\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{5}\\-\frac{3}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{18}{5},y=-\frac{3}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$7x+2y=24,-8x+2y=-30$$

Me tango -8x+2y=-30 mai i 7x+2y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x+8x+2y-2y=24+30$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7x+8x=24+30$$

Tāpiri 7x ki te 8x.

$$15x=24+30$$

Tāpiri 24 ki te 30.

$$15x=54$$

Whakawehea ngā taha e rua ki te 15.

$$x=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō x ki -8x+2y=-30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-8\times \frac{18}{5}+2y=-30$$

Whakareatia -8 ki te \frac{18}{5}.

$$-\frac{144}{5}+2y=-30$$

Me tāpiri \frac{144}{5} ki ngā taha e rua o te whārite.

$$2y=-\frac{6}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{5},y=-\frac{3}{5}$$